1 Introduction

The classic Euclidean isoperimetric inequality states that for any bounded open set \(\Omega \subset \mathbb {R}^{n+1}\) with sufficiently regular boundary it holds that

$$\begin{aligned} |\Omega | \le \gamma (n+1) |\partial \Omega |^\frac{n+1}{n} \end{aligned}$$
(1.1)

where \(|\Omega |\) is the Lebesgue measure of \(\Omega \), \(|\partial \Omega |\) is a suitable notion of measure of the boundary and \(\gamma (n+1) = (n+1)^{-(n+1)/n} \omega _{n+1}^{-1/n}\), where \(\omega _{n+1}\) is the measure of the Euclidean unit \((n+1)\)-ball. Furthermore, equality is attained if and only if \(\Omega \) is a ball.

It is a natural question if there is a corresponding statement in higher codimension. This was answered by Almgren in [Alm86], which can be loosely stated as follows.

Theorem 1.1

(Almgren). Corresponding to each m-dimensional closed surface T in \(\mathbb {R}^{n+1}\) there is an \((m+1)\)-dimensional surface Q having T as boundary such that

$$\begin{aligned} |Q| \le \gamma (m+1) |T|^\frac{m+1}{m} \end{aligned}$$
(1.2)

with equality if and only if T is a standard round m sphere (of some radius) and Q is the corresponding flat \(m+1\) disk.

Here |Q| and |T| denote the areas in dimension \(m+1\) and m respectively. To be more precise, the notion ‘surface’ can be taken to be real rectifiable currents, real polyhedral chains, integral currents or flat chains mod \(\nu \).

A further natural question to extend (1.1) is to ask on which Riemannian manifolds \((M^n,g)\) does the Euclidean isoperimetric inequality hold. A natural candidate are Cartan-Hadamard manifolds \((M^n,g)\); that is complete, simply connected Riemannian manifolds with non-positive sectional curvature. We will denote the space of such manifolds with sectional curvatures bounded from above by \(-\kappa \) for \(-\kappa \le 0\) by \({{\mathcal {C}}}{{\mathcal {H}}}(n,-\kappa )\). The following conjecture appeared in Aubin [Aub76], Burago–Zalgaller [BZ88] and Gromov [Gro81].

Conjecture

Let \((M^n,g)\in {{\mathcal {C}}}{{\mathcal {H}}}(n,0)\). Then the Euclidean isoperimetric inequality (1.1) holds on (M, g).

The conjecture can be strengthened by asking that if \((M,g) \in {{\mathcal {C}}}{{\mathcal {H}}}(n,-\kappa )\) for \(-\kappa <0\), then the isoperimetric inequality of the model space with sectional curvatures equal to \(-\kappa \) holds. This has been proven for \(n=2\) and \(\kappa =0 \) by Weil [Wei26], for \(n=2\) and \(\kappa \ne 0 \) by Bol [Bol41] , for \(n=3\) and \(-\kappa \le 0\) by Kleiner [Kle92] and for \(n=4\) and \(\kappa = 0\) by Croke [Cro84]. For a more detailed overview of the history of the conjecture and a partial extension for \(n=4\) and \(\kappa \ne 0\) see Kloeckner–Kuperberg [KK13]. The remaining cases are open. Using a variant of mean curvature flow we gave an alternative proof of Kleiner’s result in [Sch08].

The question if an isoperimetric estimate as in (1.2) holds for any dimension \(m \le n-1\) with a non-optimal constant, depending only on m, was first resolved for Euclidean space by Federer–Fleming [FF60] and by Gromov [Gro83] for a certain class of complete Riemannian manifolds, including Cartan-Hadamard manifolds. For further extensions to metric spaces see also [AK00, Gro83, Wen05].

The main result in this article is an extension of Almgren’s result to 2-dimensional surfaces in Cartan-Hadamard manifolds with arbitrary codimension.

Theorem 1.2

Let \((M^{n},g) \in {{\mathcal {C}}}{{\mathcal {H}}}(n,-\kappa ), -\kappa \le 0, n\ge 3\) and \(\Sigma \subset M\) an integral 2-current (flat chain mod 2) with compact support such that \(\partial \Sigma =0\). Let S be an area minimising integral 3-current (flat chain mod 2) such that \(\partial S = \Sigma \). Then

$$\begin{aligned} {\mathbf {M}}[\Sigma ] \ge \mathcal {H}^2(\partial B_r) , \end{aligned}$$
(1.3)

where \(B_r\) is a geodesic ball in the 3-dimensional model space with sectional curvatures equal to \(-\kappa \) and radius r such that \(\mathcal {H}^3(B_r) = {\mathbf {M}}[S]\).

Here \(\mathbf {M}[\,\cdot \,]\) denotes the mass of a current and \({\mathcal {H}}^k\) the k-dimensional Hausdorff measure in the model space. The corresponding isoperimetric inequality in a Cartan-Hadamard manifold for closed (smooth) curves bounding a smooth minimal surface in any codimension follows from general results of Reshetnyak [Res68, Res61].

We can also characterise the equality case.

Theorem 1.3

If equality in (1.3) is attained, then \(\Sigma \) is a smooth embedded 2-sphere, has unit density, its mean curvature vector has constant length and \(\Sigma \) is totally umbilic. Furthermore, it bounds a totally geodesic embedded 3-ball S, with the mean curvature vector of \(\Sigma \) proportional to the unit conormal of S at every point in \(\Sigma \). S is isometric to a geodesic ball in the 3-dimensional model space such that the mean curvature of the boundary coincides with the one of \(\Sigma \).

A central ingredient in the proof is an optimal lower bound for the Willmore energy of an integral 2-varifold with compact support.

Theorem 1.4

Let \((M^{n},g) \in {{\mathcal {C}}}{{\mathcal {H}}}(n,-\kappa ), -\kappa \le 0, n\ge 3\) and \(\Sigma ^2\) be an integral 2-varifold in M with compact support and that the first variation of \(\Sigma \) is summable in \(L^2(\mu )\). Then

$$\begin{aligned} \int _\Sigma |{\mathbf {H}}|^2 \, d\mu \ge 16\pi + 4 \kappa |\Sigma | , \end{aligned}$$
(1.4)

where \({\mathbf {H}}\) is the weak mean curvature of \(\Sigma \). If equality is attained, then \(\Sigma \) is a smooth embedded 2-sphere, has density one, the mean curvature vector has constant length and \(\Sigma \) is totally umbilic. Furthermore, it bounds a totally geodesic embedded 3-ball S, with the mean curvature vector of \(\Sigma \) proportional to the unit conormal of S at every point in \(\Sigma \). S is isometric to a geodesic ball in the 3-dimensional model space such that the mean curvature of the boundary coincides with the one of \(\Sigma \).

This estimate for \(n=3\) and \(\kappa =0\) appeared already in [Sch08, Lemma 6.7]. In Euclidean space the estimate can be found in work of Simon [Sim93], and follows rather directly from the usual calculations leading to the monotonicity formula. The characterisation of the equality case in an Euclidean ambient is given by Lamm–Schätzle [LS14] together with a stability result. For smooth surfaces in an Euclidean ambient space this is the well known Li–Yau estimate, [LY82]. For sufficiently regular surfaces in codimension one which are outward minimising, an analogous estimate following from the Gauss–Bonnet formula is central in the argument of Kleiner [Kle92] (see also the alternative proof of Ritoré [Rit05], which does not require the condition of outward minimising).

Remark 1.5

For \((M,g) \in {{\mathcal {C}}}{{\mathcal {H}}}(n,0)\) and \(\Sigma \) an integral m-varifold in M, one can use the variant of the Michael–Simon Sobolev inequality [MS73] for Riemannian manifolds by Hoffman–Spruck [HS, HS74] (which carries over to the setting of integral varifolds) to get an estimate

$$\begin{aligned} |\Sigma |^\frac{m-2}{m} \le C(m) \int _\Sigma |{\mathbf {H}}|^2 \, d\mu \end{aligned}$$

where C(m) depends only on m. This constant is not optimal, but the proof of Theorem 1.2 carries over to any dimension and codimension, yielding a non-optimal inequality for integral currents or flat chains mod 2 as in (1.2) with a constant only depending on m. Alternatively, restricting to an open, precompact set \(K \subset M\), a direct comparison with Euclidean space gives a non-optimal isoperimetric inequality, where the constant depends on (M, g) and K, see Lemma A.2.

We give a first outline of the idea of the proof of Theorem 1.2 for \(\kappa =0\). Assume \( n\ge 3, (M^n,g) \in {{\mathcal {C}}}{{\mathcal {H}}}(n,0)\) and that \(\Sigma \) is an orientable, closed, smooth, 2-dimensional submanifold of M. Let \((\Sigma _t)_{0\le t<T}\) be its smooth evolution by mean curvature flow with \(\Sigma _0 = \Sigma \). Assume further that there exists a smooth family

$$\begin{aligned} (S_t)_{0\le t<T} \end{aligned}$$
(1.5)

of minimal 3-dimensional (immersed) submanifolds in M such that \( \partial S_t = \Sigma _t\), and let \(X_t\) be the variation vectorfield along \((S_t)_{0\le t<T}\). The first variation formula then implies that

$$\begin{aligned} \frac{d}{dt} |S_t| = \int _{S_t} \text {div}_{S_t}(X)\, d\mathcal {H}^3 = \int _{\Sigma _t} \langle X, {\mathbf {n}}\rangle \, d\mathcal {H}^2 =\int _{\Sigma _t} \langle \mathbf {H}, {\mathbf {n}}\rangle \, d\mathcal {H}^2 , \end{aligned}$$

where \({\mathbf {n}}\) is the unit conormal of \(S_t\) along \(\partial S_t = \Sigma _t\) and \(\mathbf {H}\) is the mean curvature vector of \(\Sigma _t\). Similar to [Sch08], we consider the isoperimetric difference

$$\begin{aligned} I_t=|\Sigma _t|^{3/2} - 6 \sqrt{\pi } |S_t| \end{aligned}$$

and compute, using (1.4) in the second line,

$$\begin{aligned}\begin{aligned} - \frac{d}{dt} |S_t|&= - \int _{\Sigma _t} \langle \mathbf {H}, {\mathbf {n}}\rangle \, d\mathcal {H}^2 \le \int _{\Sigma _t} |\mathbf {H}| \, d\mathcal {H}^2 \\&\le \left( \int _{\Sigma _t} |\mathbf {H}|^2 \, d\mathcal {H}^2\right) ^{1/2} |\Sigma _t|^{1/2} \cdot \frac{1}{4 \sqrt{\pi }} \left( \int _{\Sigma _t} |\mathbf {H}|^2 \, d\mathcal {H}^2\right) ^{1/2} \\&\le \frac{1}{4 \sqrt{\pi }} |\Sigma _t|^{1/2}\int _{\Sigma _t} |\mathbf {H}|^2 \, d\mathcal {H}^2 = - \frac{1}{6 \sqrt{\pi }} \frac{d}{dt} |\Sigma _t|^{3/2} \end{aligned} \end{aligned}$$

and thus \(\tfrac{d}{dt} I_t \le 0\). If the flow \((\Sigma _t)_{0\le t<T}\) and the family \((S_t)_{0\le t<T}\) exists long enough such that \(\lim _{t\rightarrow T} |S_t| =0\), this shows that

$$\begin{aligned} |S_0| \le \frac{1}{6\sqrt{\pi }} |\Sigma _0|^{3/2} . \end{aligned}$$

But in general it can’t be expected that the flow does not develop singularities before the spanning volume goes zero. It is also not clear why a sufficiently regular family \((\Sigma _t)_{0\le t<T}\) should exist. To be able to evolve through singularities we would like to work with a weak solution of mean curvature flow, in our case the most suitable one seems to be a Brakke flow. But there are only very little regularity results for higher codimension, even sudden vanishing is possible. Furthermore, it is not clear to us how to construct a sufficiently regular family of spanning minimal surfaces such that the above monotonicity calculation can be performed.

To circumvent this problem we work with Ilmanen’s elliptic regularisation scheme [Ilm94]. In this work Ilmanen combines the elliptic regularisation approach of Evans–Spruck [ES91] in codimension one with the moving varifold solutions of Brakke [Bra78] to construct Brakke flow solutions with special properties. Treating all surfaces as if they were smooth and avoiding some of the technical details, we give an overview of the argument to prove Theorem 1.2 for \(\kappa = 0\).

Let \(\Sigma _0 \subset M\) be an integral 2-current with compact support such that \(\partial \Sigma _0 = 0\). We consider \(\Sigma _0 \subset M\times \{0\} \subset M \times \mathbb {R}\) and denote by z the coordinate in the additional \(\mathbb {R}\)-direction and \(\tau \) the corresponding unit vector. Ilmanen’s elliptic regularisation scheme yields a sequence \(\varepsilon _i>0,\, \varepsilon _i \rightarrow 0\), a sequence of integral 3-currents \(P^i\) such that \(\partial P^i = \Sigma _0\), which yield translating solutions to mean curvature flow in \(M \times \mathbb {R}\) via

$$\begin{aligned} P^i(t) = P^i - \frac{t}{\varepsilon _i}\tau . \end{aligned}$$

Let \(\{\mu ^i_t\}_{t \in \mathbb {R}}\) be the corresponding family of Radon measures. This sequence of flows converges as \(i \rightarrow \infty \) to a limiting Brakke flow \(\{{\bar{\mu }}_t\}_{t \ge 0}\) which is invariant in z-direction, starting at \(\Sigma _0\times \mathbb {R}\). The Brakke flow \(\{\mu _t\}_{t \ge 0}\) starting at \(\Sigma _0\) is then obtained via slicing \(\{{\bar{\mu }}^i_t\}_{t \ge 0}\) at height \(z=\text {const}\). Additionally, the sequence

$$\begin{aligned} T^i = \kappa _{\varepsilon _i}(P^i) , \end{aligned}$$

where \(\kappa _{\varepsilon _i}(x,z) = (x,\varepsilon _i z)\), converges to a current \(T \subset M\times \mathbb {R}^+\) such that \(\partial T = \Sigma _0\). Furthermore,

$$\begin{aligned} \mu _t \ge \mu _{T_t} \end{aligned}$$

where \(\mu _{T_t}\) is the mass measure associated to the slice \(T_t\) of T at height \(z=t\). The current T is called the undercurrent of the flow \(\{\mu ^i_t\}_{t \ge 0}\). Treating the z-direction as time, it can be helpful to think of T as the space-time track of the flow \(\{\mu _t\}_{t \ge 0}\), after taking into account possible cancellations. Furthermore for all \(t>0\)

$$\begin{aligned} P^i(t) \rightarrow \pi (T_t) \times \mathbb {R}\end{aligned}$$
(1.6)

as \(i \rightarrow \infty \), where \(\pi : M \times \mathbb {R}\rightarrow M\) is the projection on the first factor.

We choose \(S_0\), a mass-minimising integral 3-current with \(\partial S_0 = \Sigma \) and \(S^i\) mass-minimising integral 4-currents in \(M\times \mathbb {R}\) such that

$$\begin{aligned} \partial S^i = P^i - S_0 \end{aligned}$$

and denote

$$\begin{aligned} S^i(t) = S^i - \frac{t}{\varepsilon _i}\tau . \end{aligned}$$

This family will serve as a family of minimal surfaces approximating the family (1.5) considered in the smooth monotonicity calculation. Note that the variation vectorfield of this family is just given by \(X = - \varepsilon _i^{-1}\tau \), which makes the monotonicity calculation for (1.8) feasible.

Let \(l >1\). We choose \(\varphi _l \in C^2_c(\mathbb {R})\) such that \(0 \le \varphi _l \le 1/l\) with \(\varphi _l = 1/l\) on \([2,l+2]\), \(\varphi _l = 0\) on \((0,\infty ) \backslash [1,l+3]\). We define the approximate area and volume by

$$\begin{aligned} A^i_t = \int \varphi _l \, d\mu ^i_t \quad \text {and} \quad V^i_t:= \int \varphi _l\, d\mu ^{S,i}_t . \end{aligned}$$

The averaging function \(\varphi _l\) takes into account that in the limit \(i \rightarrow \infty ,\, P^i(t)\) becomes vertical, and thus \(A^i_t\) approximates \(\mu _t(M)\). For t fixed and \(i \rightarrow \infty \) we expect that \(S^i(t)\) has a similar behaviour and thus \(V^i_t\) approximates the measure of a family as in (1.5).

In Euclidean space shrinking spheres with radius \(R(t) = \sqrt{R^2-2mt}\) act as barriers for integral m-Brakke flows from the inside and from the outside. Using the properties of the Hessian of the distance function to a point p in a Cartan-Hadamard manifold \(M^n\) one can show that this remains true as barriers from the outside, and thus the flow \(\{\mu ^i_t\}_{t \ge 0}\) has a finite maximal existence time \(T_\text {max} \le R^2/4\), provided \(\Sigma _0 \subset B_R(p)\).

To see that

$$\begin{aligned} V^i_t < \varepsilon \text { for } t \text { close to }T_\text {max}, \, l \ge l_0\text { and }i \text { sufficiently large,} \end{aligned}$$
(1.7)

one can use the future space-time track of the flow as a competitor: motivated by the fact that an estimate for the volume traced out by a mean curvature flow is given by the \(L^1\)-norm in time of the mean curvature vector, and the natural estimate

$$\begin{aligned} \int _0^{T_\text {max}} |{\mathbf {H}}|^2 \, d\mu _t\, dt \le \mathbf {M}[\Sigma _0] , \end{aligned}$$

where \(\mathbf {M}[\Sigma _0]\) is the measure of \(\Sigma _0\), Ilmanen shows that

$$\begin{aligned} \mathbf {M}[ \pi (T\cap \{z\ge t\})] \le (T_\text {max}-t)^{1/2} \mathbf {M}[\Sigma _0]. \end{aligned}$$

Noting that \(\partial (\pi (T\cap \{z\ge t\})) = \pi (T_t)\) and recalling (1.6) we can use \(\pi (T\cap \{z\ge t\}) \times \mathbb {R}\), up to a small error, as a competitor to \(S^i(t)\) to achieve (1.7).

For the monotonicity calculation we consider the approximate isoperimetric difference

$$\begin{aligned} I^i_t=|A^i_t|^{3/2} - 6 \sqrt{\pi } |V^i_t| , \end{aligned}$$
(1.8)

and show that this quantity is monotone in the limit as \(l\rightarrow \infty \) and \(i \rightarrow \infty \) between \(t_0=0\) and \(0<t_1<T_\text {max}\). To see this we show that the error terms in the time derivative of (1.8) are controllable and combine the property that \(P^i(t)\) becomes vertical with the estimate (1.4) and the lower semicontinuity of the \(L^2\)-norm of the mean curvature. Together with (1.7) this yield that

$$\begin{aligned} (\mathbf {M}[\Sigma _0])^{3/2} \ge 6 \sqrt{\pi }\, \mathbf {M}[S_0]\, . \end{aligned}$$

Structure of the paper. In section 2 we recall Ilmanen’s elliptic regularisation scheme [Ilm94] and show the improved approximation (1.6). The barrier argument and a comparison principle due to B. White yield the estimate on the maximal existence time. We also prove a positive lower estimate on the maximal existence time for the limiting Brakke flow.

An essential ingredient in controlling the error terms when showing the almost monotonicity of the approximate isoperimetric difference is to know that

$$\begin{aligned} S^i \rightarrow S_0 \times [0,\infty )\, \end{aligned}$$
(1.9)

as \(i \rightarrow \infty \). To achieve this we first assume that \(S_0\) is the unique mass-minimising current spanning \(\Sigma _0\). Using this assumption, we show in section 3 that (1.9) holds. We later show that by perturbing \(\Sigma _0\) slightly we can assume that \(\Sigma _0\) bounds only one mass-minimising current. We also give uniform local area bounds for \(S^i\).

In section 4 we prove (1.7).

In section 5 we compute the time derivative of the approximate isoperimetric difference and show that the error terms are controllable in the limit \(i \rightarrow \infty \). We use a lower semi-continuity argument together with (1.4) to prove Theorem 1.2. We also show that we can treat the case of equality, Theorem 1.3, using the characterisation of equality in Theorem 1.4.

In section 6 we prove Theorem 1.4.

In the “Appendix” we collect several results needed in the prequel. We show that the mass minimising currents \(S^i\) are strongly stationary and that there is a non-optimal isoperimetric inequality in any dimension and codimension in a Cartan-Hadamard manifold. Furthermore, we recall White’s avoidance principle for Brakke flows and show how unique continuation for minimal surfaces in any codimension follows from work of Kazdan.

2 Elliptic Regularisation

We employ Ilmanen’s elliptic regularisation scheme [Ilm94] to construct a Brakke flow starting at \(\Sigma \). We recall the construction of Ilmanen, adapted to our setting, and its properties needed in the sequel.

Theorem 2.1

([Ilm94], §8.1). Let \(T_0\) be local integral m-current in \((M^{m+k},g)\) with \(\partial T_0 = 0\) and finite mass \(\mathbf{M}[T_0]<\infty \). Then there exists a local integral \((m+1)\)-current T in \(M\times [0,\infty )\) and a family \(\{\mu _t\}_{t\ge 0}\) of Radon measures on M such that

  1. (i)

    (a) \(\partial T = T_0\)

    (b) \(\mathbf{M}[T_B]\), where \(T_B = T\,\textsf {L}\,(M\times B),\ B\subset \mathbb {R}\), is absolutely continuous with respect to \({\mathcal {L}}^1(B)\).

  2. (ii)

    (a) \(\mu _0=\mu _{T_0}, \mathbf{M}[\mu _t]\le \mathbf{M}[\mu _0]\) for \(t>0\).

    (b) \(\{\mu _t\}_{t\ge 0}\) is an integral n-Brakke flow.

  3. (iii)

    \(\mu _t\ge \mu _{T_t}\) for each \(t\ge 0\), where \(T_t\) is the slice \(\partial (T\,\textsf {L}\,(M^{m+k}\times [t,\infty ))\).

We outline the main steps of the proof. Ilmanen constructs local integral \((m+1)\)-currents \(P_\varepsilon \) in \(M^{m+k}\times \mathbb {R}\) that minimize the elliptic translator functional

$$\begin{aligned} I^\varepsilon [Q] = \frac{1}{\varepsilon }\int e^{-z/\varepsilon }\, d\mu _Q(x,z)\, , \end{aligned}$$

where z is the coordinate in the additional \(\mathbb {R}\)-direction, subject to the boundary condition

$$\begin{aligned} \partial Q = T_0 , \end{aligned}$$

and \(M^{m+k}\) is identified with the height zero slice in \(M^{m+k}\times \mathbb {R}\). Note that \(I^\varepsilon \) is the area functional for the metric \({\bar{g}} = e^{-2z/((m+1)\varepsilon )}(g \oplus dz^2)\), where \(g \oplus dz^2\) is the product metric on \(M^{m+k}\times \mathbb {R}\).

The associated Euler–Lagrange equation implies that the family of Radon measures \(\mu ^\varepsilon _t=\mu _{P^\varepsilon _t}\) corresponding to

$$\begin{aligned} P^\varepsilon (t) = (\sigma _{-t/\varepsilon })_\# (P^\varepsilon ) \end{aligned}$$

for \(0\le t<\infty \), where \(\sigma _{-t/\varepsilon }(x,z)=(x,z-t/\varepsilon )\), is a downward translating integral \((m+1)\)-Brakke flow on the relatively open subset \(W^\varepsilon := \{(x,z,t)\, :\, z>-t/\varepsilon ,\ t\ge 0\}\) of space-time \((M^{m+k}\times \mathbb {R})\times [0,\infty )\).

Ilmanen’s compactness theorem for Brakke flows implies that there is a sequence \(\varepsilon _i\rightarrow 0\) such that \(\{\mu ^{\varepsilon _i}_t\}_{t\ge 0}\) converges to a Brakke flow \(\{{\bar{\mu }}_t\}_{t\ge 0}\) on space-time. Furthermore, Ilmanen shows that \({\bar{\mu }}_0 = \mu _{T_0\times \mathbb {R}}\) and \({\bar{\mu }}_t\) is invariant in the z-direction, which yields the desired solution \(\{\mu _t\}_{t\ge 0}\) via slicing.

The integral current T is constructed via considering a subsequential limit of \(T^{\varepsilon _i}:=(\kappa _{\varepsilon _i})_\#(P^{\varepsilon _i})\) where \(\kappa _{\varepsilon _i}(x,z)=(x,\varepsilon _i z)\), which can be seen as an approximation to the space-time track of \(\{\mu _t\}_{t\ge 0}\) where now the z-direction is considered as the time direction. Point (iii) above verifies this interpretation.

Recall that for \(s\ge 0\) we define the following slices by the height function z:

$$\begin{aligned} P^{\varepsilon _i}_s = \partial (P^{\varepsilon _i} \,\textsf {L}\,(M \times [s,\infty )) \end{aligned}$$

and similarly

$$\begin{aligned} T_s = \partial (T \,\textsf {L}\,(M \times [s,\infty )) . \end{aligned}$$

We note the following estimates from [Ilm94].

Proposition 2.2

(Ilmanen). The following estimates hold: for any measurable subset \(A \subset \mathbb {R}\)

$$\begin{aligned} {\mathbf {M}}[P^\varepsilon _A] \le (|A| + \varepsilon ) {\mathbf {M}}[T_0] \end{aligned}$$
(2.1)

where \(P_A = P\,\textsf {L}\,(M\times A)\) and |A| is the measure of A. Let \(\pi : M \times \mathbb {R}\rightarrow M\) be the projection onto M. Then for any measurable subset \(B \subset \mathbb {R}\)

$$\begin{aligned} {\mathbf {M}}(\pi _\#(T^\varepsilon _B)) \le (|B| + \varepsilon ^2)^{1/2} {\mathbf {M}}[T_0] . \end{aligned}$$
(2.2)

In particular in the flat metric distance

$$\begin{aligned} \text {dist}(\pi _\#(T^\varepsilon _t), \pi _\#(T^\varepsilon _{t+\delta })) \le (\delta + \varepsilon ^2)^{1/2} {\mathbf {M}}[T_0] . \end{aligned}$$
(2.3)

Furthermore,

$$\begin{aligned} {\mathbf {M}}[(T^\varepsilon _B)] \le \big (|B| + \varepsilon ^2+ (|B| + \varepsilon ^2)^{1/2}\big ) {\mathbf {M}}[T_0] . \end{aligned}$$
(2.4)

In particular in the flat metric distance

$$\begin{aligned} \text {{dist}}(T^\varepsilon _t, T^\varepsilon _{t+\delta }) \le \big (\delta + \varepsilon ^2+ (\delta + \varepsilon ^2)^{1/2}\big ) {\mathbf {M}}[T_0] . \end{aligned}$$
(2.5)

For details see sections 5.1–5.3 in [Ilm94].

One can use the \(C^{1/2}\)-continuity of \((T^\varepsilon _t)\) to show the following improved approximation property.

Lemma 2.3

We have

$$\begin{aligned} P^{\varepsilon _i} = P^{\varepsilon _i}(0) \rightarrow T_0\times [0,+\infty ) , \end{aligned}$$

and for \(t>0\)

$$\begin{aligned} P^{\varepsilon _i}(t) \rightarrow \pi _\#(T_t)\times \mathbb {R}\end{aligned}$$

in the sense of currents.

Proof

Fix \(t\ge 0\). By (2.1) we can assume, up to a subsequence, that \(P^{\varepsilon _i}(t) \rightarrow P'\). Recall that \(T^{\varepsilon _i} \rightarrow T\) and thus \(T^{\varepsilon _i} \,\textsf {L}\,(M \times [t, \infty )) \rightarrow T\,\textsf {L}\,(M \times [t, \infty ))\) for any \( t\ge 0\). Taking boundaries this yields

$$\begin{aligned} T^{\varepsilon _i}_t \rightarrow T_t . \end{aligned}$$

Note that \(T^{\varepsilon _i}_t = (\kappa _{\varepsilon _i})_\# P^{\varepsilon _i}_{t/\varepsilon _i}\). This implies that

$$\begin{aligned} (\kappa _{\varepsilon _i})_\# (P^{\varepsilon _i}_{t/\varepsilon _i + s}) = T^{\varepsilon _i}_{t+ \varepsilon _i s} . \end{aligned}$$

Using (2.3) this yields that for \(t=0\) and \(s\ge 0\) or \(t>0\) and any \(s\in \mathbb {R}\)

$$\begin{aligned} \pi _\#(P^{\varepsilon _i}_{t/\varepsilon _i + s}) \rightarrow \pi _\#(T_t) . \end{aligned}$$

This yields that for any any \(s\in [0,\infty )\)

$$\begin{aligned} \pi _\#(P^{\varepsilon _i}_s) \rightarrow \pi _\#(T_0) \end{aligned}$$

and thus

$$\begin{aligned} \pi _\#(P'_s) = \pi _\#(T_0) . \end{aligned}$$

Furthermore, by (2.2) we have for any \(0\le s_1 < s_2\) that

$$\begin{aligned} {\mathbf {M}}[\pi _\#(P^{\varepsilon _i}_{[s_1,s_2]})] = {\mathbf {M}}[\pi _\#(T^{\varepsilon _i}_{[\varepsilon _i s_1, \varepsilon _i s_2]})] \le (\varepsilon _i(s_2-s_1) + \varepsilon _i^2)^{1/2} {\mathbf {M}}[T_0] \end{aligned}$$

and thus

$$\begin{aligned} {\mathbf {M}}[\pi _\#(P'_{[s_1,s_2]})] = 0 . \end{aligned}$$

This yields that \(\tfrac{\partial }{\partial z}\) is \({\mathcal {H}}^{m+1}-a.e.\) tangential to \(P'\). By the coarea-formula this implies that

$$\begin{aligned} P'= \pi _\#(T_0)\times [0,\infty ) . \end{aligned}$$

For \(t>0\), we obtain that for any any \(s\in \mathbb {R}\)

$$\begin{aligned} \pi _\#((P^{\varepsilon _i}(t))_s) = \pi _\#(((\sigma _{-t/\varepsilon _i})_\#(P^{\varepsilon _i}))_s) = \pi _\#(P^{\varepsilon _i}_{t/\varepsilon _i + s}) \rightarrow \pi _\#(T_t) \end{aligned}$$

and by the same argument as earlier that

$$\begin{aligned} P^{\varepsilon _i}(t) \rightarrow P'=\pi _\#(T_t)\times \mathbb {R}. \end{aligned}$$

\(\square \)

We will in the following always assume that \((M, g) \in {{\mathcal {C}}}{{\mathcal {H}}}(n,0)\). We consider the local integral 3-currents \(P^\varepsilon \subset M\times [0,\infty )\) constructed in the previous section, such that \(\partial P^\varepsilon = \Sigma _0\). We choose a sequence \(\varepsilon _i \rightarrow 0\) such that as in the proof of Theorem 2.1, we have \(\{\mu ^{\varepsilon _i}_t\}_{t\ge 0}\) converging to a Brakke flow \(\{{\bar{\mu }}_t\}_{t\ge 0}\) which is invariant in the z-direction (which we can w.l.o.g.  assume is true for all t) and \(T^{\varepsilon _i} \rightarrow T\). Let \(\{\mu _t\}_{t\ge 0}\) be the Brakke flow starting at \(\Sigma _0\) obtained from \(\{{\bar{\mu }}_t\}_{t\ge 0}\) via slicing in z-direction. We denote the maximal existence time of the constructed Brakke flow \(\{\mu _t\}_{t\ge 0}\), by

$$\begin{aligned} T_\text {max} = \inf _{t>0}\{t\, |\, \mu _t = 0\} . \end{aligned}$$

Note that by the monotonicity of the total measure we have \(\mu _t(M)>0\) for all \(t< T_\text {max}\) and \(\mu _t(M)=0\) for all \(t>T_\text {max}\). Under the present restrictions on the geometry of M we obtain an upper bound for the maximal existence time.

Lemma 2.4

Assume \((M, g) \in {{\mathcal {C}}}{{\mathcal {H}}}(n,0)\). Let \(p_0 \in M\) and \(\mathrm{spt}\,\Sigma _0 \subset B_R(p_0)\). Then \(\mathrm{spt}\,\mu _t \subset B_{r(t)}(p_0)\) where \(r(t) = \sqrt{R^2 - 4t}\). The maximal existence time \( T_\text {max}\) of the constructed brakke flow \(\{\mu _t\}_{t\ge 0}\) is bounded from above by \(R^2/4\). Furthermore,

$$\begin{aligned} \mathrm{spt}\,P^\varepsilon \subset \big \{(p,z)\, |\, 0 \le z \le \varepsilon ^{-1} \big (R^2 + o(1) - d(p,p_0)^2\big )/4\big \} . \end{aligned}$$

Proof

Let \(r(p):= d(p,p_0)\). Since M is complete and has non-positive sectional curvature we have

$$\begin{aligned} \nabla ^2 r \ge r^{-1}(\text {id} - \nabla r \otimes \nabla r ) . \end{aligned}$$

Consider \(0<\alpha <n\) and the function

$$\begin{aligned} u(p,t) = r^2 +2\alpha t . \end{aligned}$$

Then with the notation as in Theorem A.3 we see that

$$\begin{aligned} \frac{\partial u}{\partial t} - \text {tr}_2 \nabla ^2 u = 2\alpha - 2 \text {tr}_2( r \nabla ^2r + \nabla r \otimes \nabla r) \le 2\alpha - 2 \text {tr}_2\text {id} < 0 , \end{aligned}$$

and thus by Theorem A.3

$$\begin{aligned} u(x,t) \le R^2 \end{aligned}$$

on \(\mathrm{spt}\,\mu _t\). Letting \(\alpha \rightarrow n\) this implies the first two statements. To obtain the height bound observe that by Huisken’s monotonicity formula (with a suitable local modification due to the non-flat background) the support of the Brakke flow \((\mu ^\varepsilon _t)_{t\ge 0}\) converges in Hausdorff distance to the support of \(({\bar{\mu }}_t)_{t\ge 0}\). \(\square \)

Let \(S_0\) be an area-minimising 3-current in M such that

$$\begin{aligned} \partial S_0 = T_0 . \end{aligned}$$

Note that geodesic spheres in M are convex, thus by the convex hull property we have that the support of \(S_0\) is compact. We then also obtain a lower bound on the maximal existence time.

Lemma 2.5

Assume that \(\mathbf {M}[S_0] >0\). Then there exists \(\delta = \delta (\mathbf {M}[\Sigma _0], \mathbf {M}[S_0]) >0\) and \(\eta = \eta (\mathbf {M}[\Sigma _0], \mathbf {M}[S_0])>0\) such that

$$\begin{aligned} \mu _{t}(M) \ge \eta \end{aligned}$$

for all \(0\le t < \delta \).

Proof

Let T be the undercurrent of the flow \(\{\mu _t\}_{t\ge 0}\). Note that by (2.4) we have the estimate

$$\begin{aligned} \mathbf {M}[\pi _\#(T_{[t,s]})] \le (|s-t| + |s-t|^{1/2}) {\mathbf {M}}[\Sigma _0] \end{aligned}$$

and thus for any mass-minimising integral 3-current \(S_t\) spanning \(T_t\) we can estimate

$$\begin{aligned} {\mathbf {M}}[S_t] \ge {\mathbf {M}}[S_0] - \mathbf {M}[\pi _\#(T_{[0,s]})] \ge {\mathbf {M}}[S_0] - 2\, t^{1/2} {\mathbf {M}}[\Sigma _0] \ge \frac{{\mathbf {M}}[S_0]}{2} , \end{aligned}$$

for \(t\le \delta \). By Lemma A.2 have

$$\begin{aligned} {\mathbf {M}}[T_t] \ge \eta >0 \end{aligned}$$

for all \(0\le t < \delta \) and all k sufficiently large. The claim then follows from Theorem 2.1 (iii). \(\square \)

3 Attainment of Initial Spanning Surface

We can w.l.o.g. assume that \(\mathbf {M}[S_0] >0\). We will for the moment work with the following

Assumption

We assume \(S_0\subset M\) is the unique area-minimising 3-current spanning \(\Sigma _0\).

We will later verify that in general one can perturb \(\Sigma _0\) slightly such that the uniqueness assumption is satisfied.

Let \(S^\varepsilon \) be area-minimising 4-currents in \(M\times [0,\infty )\) such that

$$\begin{aligned} \partial S^{\varepsilon _i} = P^{\varepsilon _i} - S_0 . \end{aligned}$$

In the remaining part of this section we aim to show that

$$\begin{aligned} S^{\varepsilon _i} \rightarrow S_0\times [0,\infty ) \end{aligned}$$
(3.1)

as \(\varepsilon _i \rightarrow 0\).

Lemma 3.1

Let \((M,g) \in {{\mathcal {C}}}{{\mathcal {H}}}(n,0)\). For all \(p\in M\times [0,\infty )\) and \(r \ge 1\) it holds

$$\begin{aligned} {\mathbf {M}}[S^\varepsilon \,\textsf {L}\,B_r(p)] \le \omega _{4}\Big (r^2+ \frac{\varepsilon }{3}r\Big ) {\mathbf {M}}[\Sigma ] + \frac{\omega _4}{4} {\mathbf {M}}[S_0] . \end{aligned}$$
(3.2)

Proof

The proof of the classical monotonicity formula in \(\mathbb {R}^{n}\) relies on the fact that the position vectorfield \(X(x,x_0) = x-x_0\) satisfies \(\text {div}_{T}(X) = k\), where T is an k-dimensional subspace of \(T_x\mathbb {R}^{n}\). As in the proof of Theorem 1.4 we replace the position vectorfield \(X(x,x_0)\) by

$$\begin{aligned} X_p(x) := r {\bar{\nabla }}r \end{aligned}$$

where \(r = d(x,p)\). As in the proof there, one can check that this vectorfield satisfies

$$\begin{aligned} \text {div}_{T}(X_p) \ge k , \end{aligned}$$
(3.3)

where T is an k-dimensional subspace of \(T_x ( M\times \mathbb {R})\). Using that \(S^\varepsilon \) is strongly stationary, see Lemma A.1, one obtains as in the proof of the monotonicity formula, compare [Sim83], that

$$\begin{aligned} \begin{aligned} \frac{d}{d\rho } \bigg (\frac{{\mathbf {M}}[S^\varepsilon \,\textsf {L}\,B_\rho (p)]}{\omega _4 \rho ^4} \bigg )&\ge \frac{d}{d\rho } \int _{ B_\rho (p)} \frac{|{\bar{\nabla }} ^\perp r|^2}{r^n} \, d\mu _{S^\varepsilon } + \rho ^{-5} \int _{B_\rho (p)} \langle X_p(x), {\mathbf {H}} \rangle \, d\mu _{S^\varepsilon }\\&\ \ \ - \rho ^{-5} \int _{B_\rho (p)} \langle X_p(x), {\mathbf {n}}_S \rangle \, d\mu _{\partial S^\varepsilon } . \end{aligned} \end{aligned}$$

Together with estimate (2.1) this yields for \(\rho \ge 1\) and \(0<\varepsilon <1\)

$$\begin{aligned} \begin{aligned} \frac{d}{d\rho } \bigg (\frac{{\mathbf {M}}[S^\varepsilon \,\textsf {L}\,B_\rho (p)]}{\omega _4 \rho ^4} \bigg )&\ge - \rho ^{-5} \int _{B_\rho (p)} \langle X_p(x), {\mathbf {n}}_S \rangle \, d\mu _{P^\varepsilon } - \rho ^{-5} \int _{B_\rho (p)} \langle X_p(x), {\mathbf {n}}_S \rangle \, d\mu _{S_0}\\&\ge - \rho ^{-4} ({\mathbf {M}}[P^\varepsilon \,\textsf {L}\,B_\rho (p)] + {\mathbf {M}}[S_0])\\&\ge - (2 \rho ^{-3}+ \varepsilon \rho ^{-4}) {\mathbf {M}}[\Sigma ] - \rho ^{-4} {\mathbf {M}}[S_0] . \end{aligned} \end{aligned}$$

Integrating this for \(1\le r<R\) from r to R yields

$$\begin{aligned} \begin{aligned} \frac{{\mathbf {M}}[S^\varepsilon \,\textsf {L}\,B_r(p)]}{\omega _4 r^4}&\le \Theta (S^\varepsilon ,p,R) + (r^{-2} - R^{-2}){\mathbf {M}}[\Sigma ] + \varepsilon \frac{1}{3} (r^{-3} - R^{-3}){\mathbf {M}}[\Sigma ]\\&\quad + \frac{1}{4} (r^{-4} - R^{-4}) {\mathbf {M}}[S_0] . \end{aligned} \end{aligned}$$

Letting \(R\rightarrow \infty \) yields the desired estimate. \(\square \)

By the uniform local area bound we can thus, up to a subsequence, assume that

$$\begin{aligned} S^{\varepsilon _i} \rightarrow S' \end{aligned}$$

where \(S'\) is locally mass-minimising and satisfies

$$\begin{aligned} \partial S' = \Sigma _0\times [0,\infty ) - S_0 . \end{aligned}$$

We will define for a general integral current Q its slice at height t by

$$\begin{aligned} Q_t = \partial (Q \,\textsf {L}\,(M\times (t, \infty )) - (\partial Q)\,\textsf {L}\,(t,\infty ) \end{aligned}$$

which is compatible with the convention used by Ilmanen in [Ilm94].

Lemma 3.2

We have \(S' = S_0 \times [0,\infty )\).

Proof

We consider for \(t>0\) the slice \(S'_t\) of \(S'\) at height t as above.

Claim 1

There exists a sequence \(t_j \rightarrow \infty \) and \(C>0\) such that

$$\begin{aligned} {\mathbf {M}}[S'_{t_j}]\le C . \end{aligned}$$

This follows since by the coarea formula, and the locally uniform area estimates

$$\begin{aligned} \int _{t}^{t+1} {\mathbf {M}}[S'_\tau ]\, d\tau \le {\mathbf {M}}[S'\,\textsf {L}\,\{t\le z \le t+1\}] \le C < \infty \end{aligned}$$

independent of \(t>0\).

Claim 2

There exists a sequence \(t'_l \rightarrow \infty \) such that

$$\begin{aligned} {\mathbf {M}}[S'_{t'_l}] \rightarrow {\mathbf {M}} [S_0]. \end{aligned}$$

Note that \(\partial S'_{t_j} = \partial ((\sigma _{t_j})_{\#}(S_0)) = (\sigma _{t_j})_{\#}(\Sigma _0)\). By Claim 1 and Lemma A.2 there exists \(T_j\) such that \(\partial T_j = (\sigma _{t_j})_{\#}(S_0) - S'_{t_j}\) and \(C'>0\) such that

$$\begin{aligned} {\mathbf {M}}(T_j) \le C' \end{aligned}$$

independent of j. Assume that there exists a \(\delta >0\) such that

$$\begin{aligned} {\mathbf {M}}[S'_{t}] \ge {\mathbf {M}} [S_0] + \delta \end{aligned}$$

for all \(t>t_0\) for \(t_0\) sufficiently large. Since both \(S'\) and \(S_0\times [0,\infty )\) are locally mass minimising we obtain

$$\begin{aligned} {\mathbf {M}}[S'\,\textsf {L}\,\{ 0 \le z \le t_j\} ]\le t_j\, {\mathbf {M}}[S_0] + C' . \end{aligned}$$

But then by the coarea formula

$$\begin{aligned}&(t_j-t_0)({\mathbf {M}}(S_0) +\delta ) + t_0 {\mathbf {M}}(S_0) \\&\quad \le \int _0^{t_j} {\mathbf {M}}[S'_{\tau }]\, d\tau \le {\mathbf {M}}[S'\,\textsf {L}\,\{ 0 \le z \le t_j\} ]\le t_j\, {\mathbf {M}}[S_0] + C' , \end{aligned}$$

which yields a contradiction for j sufficiently large.

Since by assumption \(S_0\) is the unique mass-minimising current spanning \(\Sigma _0\) we obtain that

$$\begin{aligned} (\sigma _{-t'_l})_\#(S_{t'_l}) \rightarrow S_0 \end{aligned}$$

in flat norm and in mass. Thus there exists a sequence \(\delta _l \rightarrow 0\) such that

$$\begin{aligned} {\mathbf {M}}[S'\,\textsf {L}\,\{ 0 \le z \le t_j\} ]\le t_j\, {\mathbf {M}}[S_0] + \delta _l . \end{aligned}$$

But again this yields

$$\begin{aligned} \int _0^{t'_l} {\mathbf {M}}[S'_{\tau }]\, d\tau \le {\mathbf {M}}[S'\,\textsf {L}\,\{ 0 \le z \le t_j\} ]\le t'_l\, {\mathbf {M}}[S_0] + \delta _l , \end{aligned}$$

which implies

$$\begin{aligned} \int _0^{t'_j} {\mathbf {M}}[S'_{\tau }] - {\mathbf {M}}[S_0]\, d\tau \le \delta _l \end{aligned}$$

and thus in the limit \(l\rightarrow \infty \) that

$$\begin{aligned} {\mathbf {M}}[S'_t] = {\mathbf {M}}[S_0] \end{aligned}$$

and \(S'_t = (\sigma _t)_\#(S_0)\) for all \(t>0\). \(\square \)

4 Vanishing of the Spanning Area at the Final Time

For convenience of notation we will in the following replace a sub- or superscript \(\varepsilon _i\) by i.

Let \(l >1\). We choose \(\varphi _l \in C^2_c(\mathbb {R})\) such that \(0 \le \varphi _l \le 1/l\) with \(\varphi _l = 1/l\) on \([2,l+2]\), \(\varphi _l = 0\) on \((0,\infty ) \backslash [1,l+3]\) and \(|D\varphi _l|, |D^2 \varphi _l| \le 2/l\).

Recall that \(\mu ^i_t\) is the associated Radon measure of \(P^{\varepsilon _i}(t)\). We denote with \(\mu ^{S,i}_t\) the associated Radon measure of

$$\begin{aligned} S^i(t) = (\sigma _{-t/{\varepsilon _i}})_\#(S^{\varepsilon _i}) . \end{aligned}$$

We define the approximate volume by

$$\begin{aligned} V^i_t:= \int \varphi _l\, d\mu ^{S,i}_t . \end{aligned}$$

We show that the approximate volume goes to zero as \(t\rightarrow T_\text {max}\).

Lemma 4.1

There exists a constant \(C>0\) such that the following holds. Let \(0<t<T_\text {max}\) and \(l>1\). Then for i sufficiently large

$$\begin{aligned} {\mathbf {M}}[S^i(t) \,\textsf {L}\,(M\times [2,2+l])] \le (1+l) |T_\text {max}-t|^{1/2} {\mathbf {M}}[\Sigma _0] + C . \end{aligned}$$

Proof

We use T to construct a competitor to \(S^i(t)\). Recall that

$$\begin{aligned} \partial S^i(t) = P^i(t) + (\sigma _{-t/{\varepsilon _i}})_\#(S_0) \end{aligned}$$

and by Lemma 2.3 that

$$\begin{aligned} P^i(t) \rightarrow \pi _\#(T_t)\times \mathbb {R}. \end{aligned}$$
(4.1)

Let \({\bar{T}} = \pi _\#(T_{[t, \infty )})\). Note further that

$$\begin{aligned} \partial {\bar{T}} = \partial (\pi _\#(T_{[t, \infty )})) = \pi _\#(T_t) \end{aligned}$$

and since \(\mathrm{spt}\,T \subset M \times [0,T_\text {max}]\) by (2.2)

$$\begin{aligned} {\mathbf {M}}[\pi _\#(T_{[t,\infty )})] \le (T_\text {max}-t )^{1/2}{\mathbf {M}}[\Sigma _0] . \end{aligned}$$

By the co-area formula (as in the proof of claim 1 in the proof of Lemma 3.2), there exists a \(C>0\) and \(\eta _i \in [0,1]\) such that for all i

$$\begin{aligned} \mathbf {M}[S^i_{1+\eta _i}] + \mathbf {M}[S^i_{2+l+\eta _i}] \le C . \end{aligned}$$
(4.2)

Note further that

$$\begin{aligned} \pi _\#(P^i(t)\,\textsf {L}\,[1+\eta _i,\infty )) = \pi _\#(T^i_{[t+(1+\eta _i)\varepsilon _i,\infty )}) \rightarrow {\bar{T}} \end{aligned}$$

as well as

$$\begin{aligned} \pi _\#(P^i(t)\,\textsf {L}\,[2+l+\eta _i,\infty )) = \pi _\#(T^i_{[t+(2+l+\eta _i)\varepsilon _i,\infty )}) \rightarrow {\bar{T}} . \end{aligned}$$

Consider

$$\begin{aligned} R^i_-= (\sigma _{1+\eta _i})_\#(\pi _\#(P^i(t)\,\textsf {L}\,(M \times [1+\eta _i,\infty ))) \rightarrow ({\bar{T}}\times \mathbb {R})_{1+\eta _i} \end{aligned}$$

and

$$\begin{aligned} R^i_+= (\sigma _{2+l+\eta _i})_\#(\pi _\#(P^i(t)\,\textsf {L}\,(M\times [2+l+\eta _i,\infty ))) \rightarrow ({\bar{T}}\times \mathbb {R})_{2+l+\eta _i} \end{aligned}$$

and note that

$$\begin{aligned} \partial ((S^i(t))_{1+\eta _i}) = \partial R^i_- \ \ \text {and}\ \ \partial ((S^i(t))_{2+l+\eta _i}) = \partial R^i_+ . \end{aligned}$$

By the uniform mass bounds on \(S^i_{1+\eta _i},S^i_{2+l+\eta _i},R^i_-, R^i_+\) given by (4.2), (3.2) together with Lemmas 2.4 and A.2 there exits \(D^i_-, D^i_+\) such that

$$\begin{aligned} \partial D^i_- = R^i_- - S^i_2 \ \ \text {and}\ \ \partial D^i_+ = R^i_+ - S^i_{2+l} , \end{aligned}$$

and a constant C such that

$$\begin{aligned} {\mathbf {M}}[D^i_\pm ] \le C. \end{aligned}$$

We can now assume that \(\eta _i \rightarrow \eta \in [0,1]\), and thus note

$$\begin{aligned} P^i(t)\,\textsf {L}\,(M\times [1+\eta _i,l+2+\eta _i]) + R^i_ - -R^i_+ \rightarrow \partial (({\bar{T}}\times \mathbb {R}) \,\textsf {L}\,(M\times [1+\eta ,l+2+\eta ])) , \end{aligned}$$

and thus by equivalence to convergence in the flat norm, there exits \(Q^i\) such that

$$\begin{aligned} \partial Q^i= & {} ( P^i(t)\,\textsf {L}\,(M\times [1+\eta _,l+2+\eta _i]) + R^i_ - -R^i_+) \\&- \partial (({\bar{T}}\times \mathbb {R}) \,\textsf {L}\,(M\times [1+\eta ,l+2+\eta ])) \end{aligned}$$

and \({\mathbf {M}}[Q_i] \rightarrow 0\). Since \(S^i\) is locally mass minimising we can use

$$\begin{aligned} (({\bar{T}}\times \mathbb {R}) \,\textsf {L}\,(M\times [1+\eta ,l+2+\eta ])) + Q_i -D^i_-+ D^i_+ \end{aligned}$$

as a competitor to get the desired estimate. \(\square \)

Corollary 4.2

For every \(\varepsilon >0\) there exists \(l_0>0, \delta >0\) such that if \(l\ge l_0\), \(T_\text {max}-\delta<t< T_\text {max}\), then

$$\begin{aligned} V^i_t \le \varepsilon \end{aligned}$$

for i sufficiently large.

Proof

We can estimate, using Lemma 4.1, that for i sufficiently large

$$\begin{aligned} \begin{aligned} V^i_t= \int \varphi _l\, d\mu ^{S,i}_t&\le 1/l\, {\mathbf {M}}[S^i(t) \,\textsf {L}\,(M\times [2,2+l])] + 1/l\, {\mathbf {M}}[S^i(t) \,\textsf {L}\,(M\times [1,2])]\\&\ \ \ + 1/l \, {\mathbf {M}}[S^i(t) \,\textsf {L}\,(M\times [l+2, l+3])] \\&\le 2\delta ^{1/2} {\mathbf {M}}[\Sigma _0] + C/l < \varepsilon . \end{aligned} \end{aligned}$$

\(\square \)

5 The Monotonicity Calculation

Recall the approximate volume

$$\begin{aligned} V^i_t:= \int \varphi _l\, d\mu ^{S,i}_t , \end{aligned}$$

where \(|\nabla \varphi _l|, |\nabla ^2 \varphi _l| \le 2/l\). Note that we can further assume that

$$\begin{aligned} \frac{|\nabla \varphi _l|^2}{\varphi _l} \le C/l . \end{aligned}$$
(5.1)

We define the approximate area as

$$\begin{aligned} A^i_t:= \int \varphi _l \, d\mu ^i_t . \end{aligned}$$

We compute

$$\begin{aligned} \begin{aligned} \frac{d}{dt}V^i_t&= \frac{d}{dt} \int \varphi _l d\mu ^{S,i}_t = \int \langle \nabla \varphi _l, - \varepsilon ^{-1}\tau \rangle + \varphi _l\, \text {div}( - \varepsilon ^{-1}\tau )\, d\mu ^{S,i}_t\\&= - \int \langle \nabla \varphi _l, \varepsilon ^{-1}\tau ^\perp \rangle + \text {div}(\varphi _l \varepsilon ^{-1}\tau )\, d\mu ^{S,i}_t \\&= - \int \langle \nabla \varphi _l, \varepsilon ^{-1}\tau ^\perp \rangle \, d\mu ^{S,i}_t- \int \varphi _l \, \langle \varepsilon ^{-1}\tau , {\mathbf {n}}\rangle \, d\mu ^i_t \\&= - \int \langle \nabla \varphi _l, \varepsilon ^{-1}\tau ^\perp \rangle \, d\mu ^{S,i}_t + \int \varphi _l\, \langle {\mathbf {H}}, {\mathbf {n}}\rangle \, d\mu ^i_t \end{aligned} \end{aligned}$$

where we used Lemma A.1 in the step from the second to the third line. Following an idea of Huisken–Ilmanen [HI01, Lemma 5.3], we can rewrite the first term as a derivative:

$$\begin{aligned} \begin{aligned} \int \langle \nabla \varphi _l, \varepsilon ^{-1}\tau ^\perp \rangle \, d\mu ^{S,i}_t&= \int \varepsilon ^{-1} \varphi _l'(z) \langle \tau , \tau ^\perp \rangle \, d\mu ^{S,i}_t\\&= \int \varepsilon ^{-1} \varphi '_l(z-\varepsilon ^{-1}t) \langle \tau , \tau ^\perp \rangle \, d\mu ^{S,i}_0 \\&= - \frac{d}{dt} \int \varphi _l(z-\varepsilon ^{-1}t) \langle \tau , \tau ^\perp \rangle \, d\mu ^{S,i}_0\\&= - \frac{d}{dt} \int \varphi _l(z) \langle \tau , \tau ^\perp \rangle \, d\mu ^{S,i}_t . \end{aligned} \end{aligned}$$

Thus we get for \(0\le t_1 < t_2\) that

$$\begin{aligned} \begin{aligned} V^i_{t_2}-V^i_{t_1}&= \int _{t_1}^{t_2} \int \varphi _l\, \langle {\mathbf {H}}, {\mathbf {n}}\rangle \, d\mu ^i_t\, dt \\&\ \ \ + \int \varphi _l\, \langle \tau , \tau ^\perp \rangle \, d\mu ^{S,i}_{t_2} - \int \varphi _l\, \langle \tau , \tau ^\perp \rangle \, d\mu ^{S,i}_{t_1}. \end{aligned} \end{aligned}$$
(5.2)

For the approximate area we get, using (5.1),

$$\begin{aligned} \begin{aligned} \frac{d}{dt} A^i_t = \int \langle \nabla \varphi _l, {\mathbf {H}}\rangle -\varphi _l |{\mathbf {H}}|^2\, d\mu ^i_t&\le \int _{\{\nabla \varphi _l \ne 0\}} \frac{|\nabla \varphi _l|^2}{\varphi _l} d\mu ^i_t - \frac{1}{2}\int \varphi _l |{\mathbf {H}}|^2\, d\mu ^i_t\\&\le \frac{C}{l} - \frac{1}{2}\int \varphi _l |{\mathbf {H}}|^2\, d\mu ^i_t \end{aligned} \end{aligned}$$

where we used the uniform local area bounds to estimate the first integral on the right hand side. This implies the estimate

$$\begin{aligned} A^i_{t_2} + \frac{1}{2} \int _{t_1}^{t_2}\int \varphi _l |{\mathbf {H}}|^2\, d\mu ^i_t\, dt \le A^i_{t_1} + \frac{C}{l} (t_2-t_1). \end{aligned}$$
(5.3)

Note that by (2.1) this implies that

$$\begin{aligned} A^i_t \le 2 {\mathbf {M}}[\Sigma _0] + \frac{C}{l} t. \end{aligned}$$
(5.4)

We define the function \(f_\kappa :\mathbb {R}^+\rightarrow \mathbb {R}^+\) by

$$\begin{aligned} f_\kappa (A):=\int _0^A\frac{a^\frac{1}{2}}{(16\pi + 4\kappa a)^\frac{1}{2}}\, da . \end{aligned}$$

Note that \(f_0 = \tfrac{1}{6 \sqrt{\pi }} a^{3/2}\).

Let \((M_t)_{0\le t <T}\) be a smooth mean curvature flow of closed, embedded hypersurfaces in a Cartan-Hadamard manifold \((M^3,g)\) with sectional curvatures bounded above by \(-\kappa \). Let V(t) be the enclosed volume and A(t) the area. We then can apply Theorem 1.4 to estimate

$$\begin{aligned} \begin{aligned} -\frac{d}{dt}V =&\ \int _{M_t} H\, d\mathcal {H}^2 \le \bigg ( \int _{M_t} H^{2}\, d\mathcal {H}^2\bigg )^\frac{1}{2} A^\frac{1}{2}\\&\ \cdot (16\pi +4\kappa A)^{-\frac{1}{2}}\bigg ( \int _{M_t} H^{2}\, d\mathcal {H}^2\bigg )^\frac{1}{2} \\ =&\ (16\pi +4\kappa A)^{-\frac{1}{2}}A^\frac{1}{2}\int _{M_t} H^{2}\, d\mathcal {H}^2 = -\frac{d}{dt}f_\kappa (A) . \end{aligned} \end{aligned}$$

Thus \(f_\kappa (A)-V\) is monotonically decreasing under the flow. Consider the case that \(M_\kappa ^3\) is the model space of constant curvature \(-\kappa \) and let \(M_t\) be the mean curvature flow of geodesic spheres contracting to a point. Then the estimate of Theorem 1.4 holds with equality for all \(M_t\) and also the above calculation is an equality. Using that in the model space geodesic balls optimize the isoperimetric ratio, we have

$$\begin{aligned} {\mathcal {H}}^3(U) \le f_\kappa (\mathcal {H}^2(\partial U)) , \end{aligned}$$

for all open and bounded \(U\subset M^3_\kappa \), with equality on geodesic balls.

We consider the approximate isoperimetric difference

$$\begin{aligned} I^i_t = f_\kappa (A^i_t) - V^i_t , \end{aligned}$$

and compute

$$\begin{aligned} \begin{aligned} f_\kappa (A^i_{t_2}) - f_\kappa (A^i_{t_1}) =&- \int _{t_1}^{t_2} \frac{(A^i_t)^{1/2}}{(16\pi + 4\kappa A^i_t)^\frac{1}{2}} \int \varphi _l |{\mathbf {H}}|^2 \, d\mu ^i_t\, dt\\&+\int _{t_1}^{t_2} \frac{(A^i_t)^{1/2}}{(16\pi + 4\kappa A^i_t)^\frac{1}{2}} \int \langle \nabla \varphi _l, {\mathbf {H}}\rangle \, d\mu ^i_t\, dt \end{aligned} \end{aligned}$$

where we can estimate, assuming \(t_2 \le T_\text {max}\) and using (5.3), (5.4)

$$\begin{aligned} \begin{aligned} \left| \int _{t_1}^{t_2} \frac{(A^i_t)^{1/2}}{(16\pi + 4\kappa A^i_t)^\frac{1}{2}} \int \langle \nabla \varphi _l, {\mathbf {H}}\rangle \, d\mu ^i_t\, dt \right|&\le \left| \int _{t_1}^{t_2} \frac{(A^i_t)^{1/2}}{(16\pi + 4\kappa A^i_t)^\frac{1}{2}} \int |\nabla \varphi _l| |{\mathbf {H}}| \, d\mu ^i_t\, dt \right| \\&\le C \left( \int _{t_1}^{t_2} \int _{\{\nabla \varphi _l \ne 0\}} \frac{|D\varphi |^2}{\varphi } \, d\mu ^i_t\, dt \right) ^{1/2}\\&\qquad \left( \int _{t_1}^{t_2} \int \varphi _l |{\mathbf {H}}|^2 \, d\mu ^i_t\, dt \right) ^{1/2}\\&\le \frac{C}{l^{1/2}}(t_2-t_1) . \end{aligned} \end{aligned}$$

This yields the estimate

$$\begin{aligned} \begin{aligned} f_\kappa (A^i_{t_2}) - f_\kappa (A^i_{t_1})&\le - \int _{t_1}^{t_2} \frac{(A^i_t)^{1/2}}{(16\pi + 4\kappa A^i_t)^\frac{1}{2}} \int \varphi _l |{\mathbf {H}}|^2 \, d\mu ^i_t\, dt \\&\qquad + \frac{C}{l^{1/2}}(t_2-t_1) \end{aligned} \end{aligned}$$
(5.5)

for \(0\le t_1 <t_2 \le T_\text {max}\).

From (5.2), (5.5) we get the estimate

$$\begin{aligned} \begin{aligned} I^i_{t_2}-I^i_{t_1}&\le - \int _{t_1}^{t_2} \frac{(A^i_t)^{1/2}}{(16\pi + 4\kappa A^i_t)^\frac{1}{2}} \int \varphi _l |{\mathbf {H}}|^2 \, d\mu ^i_t\, dt + \int _{t_1}^{t_2} \int \varphi _l |{\mathbf {H}}|\, d\mu ^i_t\, dt \\&\ \ \ + \int \varphi _l\, \langle \tau , \tau ^\perp \rangle \, d\mu ^{S,i}_{t_1} - \int \varphi _l\, \langle \tau , \tau ^\perp \rangle \, d\mu ^{S,i}_{t_2} + \frac{C}{l^{1/2}}(t_2-t_1)\\&\le - \int _{t_1}^{t_2} \frac{(A^i_t)^{1/2}}{(16\pi + 4\kappa A^i_t)^\frac{1}{2}} \int \varphi _l |{\mathbf {H}}|^2 \, d\mu ^i_t\, dt\\&\ \ \ + \int _{t_1}^{t_2} \left( \int \varphi _l |{\mathbf {H}}|^2\, d\mu ^i_t\right) ^{1/2} (A^i_t)^{1/2}\, dt \\&\ \ \ + \int \varphi _l\, \langle \tau , \tau ^\perp \rangle \, d\mu ^{S,i}_{t_1} - \int \varphi _l\, \langle \tau , \tau ^\perp \rangle \, d\mu ^{S,i}_{t_2} + \frac{C}{l^{1/2}}(t_2-t_1) \end{aligned} \end{aligned}$$
(5.6)

for \(0\le t_1 <t_2 \le T_\text {max}\).

Recall that the limiting Brakke flow \(({\bar{\mu }}_t)_{(0\le t\le T_\text {max})}\) is invariant in the z-direction, and for a.e. t the measure \({\bar{\mu }}_t\) is 3-rectifiable and carries a weak mean curvature in \(L^2\). Using Theorem 1.4 we thus see that

$$\begin{aligned} \int \varphi _l |{\mathbf {H}}|^2 \, d{\bar{\mu }}_t \ge 16\pi + 4 \kappa \, {\bar{\mu }}_t(\varphi _l) \end{aligned}$$
(5.7)

for a.e. \(t \in [0,T_\text {max}]\).

Lemma 5.1

For any \(t_1,t_2 \in [0,T_\text {max})\), \(t_1<t_2\),

$$\begin{aligned} \limsup _{i\rightarrow \infty } \int _{t_1}^{t_2}L^i_t\, dt\le \int _{t_1}^{t_2}L_t\, dt \le 0 , \end{aligned}$$

where

$$\begin{aligned} L^i_t:= \big (A^i_t\big )^\frac{1}{2}\bigg (\bigg (\int \varphi _l |{\mathbf {H}}|^2\, d\mu ^i_t\bigg )^\frac{1}{2} - (16\pi + 4\kappa A^i_t)^{-\frac{1}{2}} \int \varphi _l |{\mathbf {H}}|^{2}\, d\mu ^i_t\bigg ) \end{aligned}$$

and

$$\begin{aligned} L_t:=\big ({\bar{\mu }}_t(\varphi _l)\big )^\frac{1}{2}\bigg ( \bigg (\int \varphi _l |{\mathbf {H}}|^2\, d{\bar{\mu }}_t\bigg )^\frac{1}{2} - (16\pi + 4\kappa {\bar{\mu }}_t(\varphi _l))^{-\frac{1}{2}} \int \varphi _l |{\mathbf {H}}|^{2}\, d{\bar{\mu }}_t\bigg ). \end{aligned}$$

Proof

From Ilmanen’s compactness theorem for Brakke flows, we know that for all \(t \in [t_1,t_2]\) we have

$$\begin{aligned} A^i_t \rightarrow {\bar{\mu }}_t(\varphi _l) , \end{aligned}$$

and by the lower semicontinuity of the \(L^2\) norm of \({\mathbf {H}}\) and (5.7) that

$$\begin{aligned} \liminf _{i \rightarrow \infty } \int \varphi _l |{\mathbf {H}}|^2\, d\mu ^i_t \ge \int \varphi _l |{\mathbf {H}}|^2\, d{\bar{\mu }}_t \ge 16\pi + 4 \kappa \, {\bar{\mu }}_t(\varphi _l) . \end{aligned}$$
(5.8)

We write \(L^i_t\) in the form \(L^i_t= a_i \cdot b_i\), where

$$\begin{aligned} a_i(t):=\big (A^i_t\big )^\frac{1}{2} \end{aligned}$$

and

$$\begin{aligned} b_i(t):= \bigg (\int \varphi _l |{\mathbf {H}}|^2\, d\mu ^i_t\bigg )^\frac{1}{2} - (16\pi + 4\kappa A^i_t)^{-\frac{1}{2}} \int \varphi _l |{\mathbf {H}}|^2\, d\mu ^i_t. \end{aligned}$$

Since the function \(x^{1/2} - (16 \pi + 4\kappa a)^{-\frac{1}{2}} x\) is decreasing for \(x\ge 4\pi + \kappa a\) we obtain, using (5.8), that

$$\begin{aligned} \limsup _{i \rightarrow \infty } b_i(t) \le \bigg (\int \varphi _l |{\mathbf {H}}|^2\, d{\bar{\mu }}_t\bigg )^\frac{1}{2} - (16\pi + 4 \kappa {\bar{\mu }}_t(\varphi _l) )^{-\frac{1}{2}} \int \varphi _l |{\mathbf {H}}|^2\, d{\bar{\mu }}_t \le 0 \end{aligned}$$

for all \(t \in [t_1,t_2]\). Together with \(a_i(t)\rightarrow \big ({\bar{\mu }}_t(\varphi _l)\big )^{1/2}\) this implies that

$$\begin{aligned} \limsup L^i_t \le L_t \le 0 \quad \text {for all } t \in [t_1,t_2] . \end{aligned}$$

Note that by (5.4) there is \(C\ge 0\) such that

$$\begin{aligned} L^i_t \le C \end{aligned}$$

for all \(t\in [t_1,t_2]\) and all \(i\ge i_0\). Then the claim follows from Fatou’s lemma. \(\square \)

We will now explain how to perturb \(\Sigma \) slightly such that we can assume that the mass minimising 3-current spanning \(\Sigma \) is unique. Let S be any mass minimising 3-current spanning \(\Sigma \). Note that by Almgren [Alm00], see also De Lellis–Spadaro [LS11, LS15, LS14, LS16, LS16], the interior singular set of S has codimension 2. Note further that the interior regular set \({\mathcal {R}}_\text {int}\) can have at most countably connected components, since S has finite mass. We denote these components by \(R_j\) for \(j \in 1,\ldots , N\) where \(N \in \mathbb {N}\cup \{\infty \}\). We can pick points \(p_j \in R_j\) and radii \(r_j> 0\) such that

  • \(B_{r_j}(p_j) \cap \mathrm{spt}\,\Sigma = \emptyset \),

  • the balls \(B_{r_j}(p_j)\) are pairwise disjoint,

  • \(S \,\textsf {L}\,B_{r_j}(p_j)\) is smooth and consists of one single, smooth, connected component,

  • \(\sum _{i=1}^N {\mathbf {M}}[S\,\textsf {L}\,B_{r_j}(p_j)] \le 1\).

For \(k \in \mathbb {N}\) let \(S_k = S \backslash \bigcup _{j=1}^N B_{r_j/k}(p_j)\) and

$$\begin{aligned} \Sigma _k := \partial S_k. \end{aligned}$$

The above estimates yield that

$$\begin{aligned} \Sigma _k \rightarrow \Sigma \end{aligned}$$
(5.9)

in flat norm and in mass.

Lemma 5.2

\(S_k\) is the unique mass minimizing current spanning \(\Sigma _k\).

Proof

Assume there is another mass minimizing current \(S'\) spanning \(\Sigma _k\). But then

$$\begin{aligned} S'':= \displaystyle S' + \bigcup _{j=1}^N S\,\textsf {L}\,{\overline{B}}_{r_j/k}(p_j) \end{aligned}$$

is mass minimising and bounds \(\Sigma \). The interior singular set of \(S''\) has again codimension 2. Thus by unique continuation, see section A.4, \(S''\) has to coincide with S on each connected component \(R_j\) of \({\mathcal {R}}_\text {int}\). Note that \(S''\) can have no further connected components of its interior regular set, since it has the same mass as S. Thus \(S'' = S\) and \(S'=S_k\). \(\square \)

Proof of Theorem 1.2

We will present the proof in the case \(\Sigma \) is an integral 2-current. The necessary modifications if \(\Sigma \) is a 2-dimensional flat chain mod 2 will be discussed at the end of the proof.

We first replace \(\Sigma \) by \(\Sigma _k\) such that by Lemma 5.2\(S_k\) is the unique area minimising current spanned by \(\Sigma _k\).

As outlined above we use Ilmanen’s elliptic regularisation scheme to construct Brakke flows \((\mu ^k_t)_{(0\le t\le T^k_\text {max})}\), starting at \(\Sigma _{k,0}:= \Sigma _k\), which vanish at a finite time \(T^k_\text {max}\). These flows arise as the slice of the translation invariant flows \(({\bar{\mu }}^k_t)_{(0\le t\le T^k_\text {max})}\) on \(M \times \mathbb {R}\), obtained as a limit of approximating flows \((\mu ^{k,i}_t)_{t \ge 0}\).

We will for the moment omit the index k. We use the set-up as before. Note that by Lemma 3.2 we have that

$$\begin{aligned} S^i(0) = S^i \rightarrow S \times [0,\infty ). \end{aligned}$$
(5.10)

Given \(\varepsilon >0\), we choose \(t_1=0\) and \(T_\text {max} -\delta< t_2 < T_\text {max}\), \(l >l_0\), where \(\delta >0\) and \(l_0\) are given by Corollary 4.2. By (5.6) and Lemma 5.1 we can estimate

$$\begin{aligned} \begin{aligned} \limsup _{i\rightarrow \infty } (I^i_{t_2}-I^i_{0})&\le \limsup _{i\rightarrow \infty } \left| \int \varphi _l\, \langle \tau , \tau ^\perp \rangle \, d\mu ^{S,i}_{0}\right| \\&\quad + \limsup _{i\rightarrow \infty }\left| \int \varphi _l\, \langle \tau , \tau ^\perp \rangle \, d\mu ^{S,i}_{t_2}\right| + \frac{C}{l^{1/2}}T_\text {max}. \end{aligned} \end{aligned}$$
(5.11)

Note that by (5.10) we have that

$$\begin{aligned} \lim _{i\rightarrow \infty } \int \varphi _l\, \langle \tau , \tau ^\perp \rangle \, d\mu ^{S,i}_{0} = 0 \end{aligned}$$

and by Corollary 4.2

$$\begin{aligned} \limsup _{i\rightarrow \infty }\left| \int \varphi _l\, \langle \tau , \tau ^\perp \rangle \, d\mu ^{S,i}_{t_2}\right| \le \varepsilon . \end{aligned}$$

Furthermore

$$\begin{aligned} \lim _{i\rightarrow \infty } I^i_{0} = f_\kappa \left( \int _{\Sigma _k \times \mathbb {R}} \varphi _l \, d\mathcal {H}^3\right) - \int _{S_k \times \mathbb {R}} \varphi _l \, d\mathcal {H}^4 . \end{aligned}$$

Again by Corollary 4.2 we have

$$\begin{aligned} \limsup _{i\rightarrow \infty } I^i_{t_2} \ge - \varepsilon . \end{aligned}$$

Putting this together we obtain

$$\begin{aligned} f_\kappa \left( \int _{\Sigma _k \times \mathbb {R}} \varphi _l \, d\mathcal {H}^3\right) - \int _{S_k \times \mathbb {R}} \varphi _l \, d\mathcal {H}^4 \ge - C \varepsilon - \frac{C}{l^{1/2}} . \end{aligned}$$

Letting first \(l \rightarrow \infty \) and then \(\varepsilon \rightarrow 0\) yields the isoperimetric inequality for \(\Sigma _k\). We can now let \(k \rightarrow \infty \) to obtain the isoperimetric inequality for \(\Sigma \).

The equality case: In case of equality in (1.3) we have

$$\begin{aligned} f_\kappa \left( {\mathbf {M}}[\Sigma _k]\right) - {\mathbf {M}}[S_k] \rightarrow 0 \end{aligned}$$

and the monoticity calculation, using Lemma 5.1, yields

$$\begin{aligned} \int _0^{T_{\text {max},k}} \big (\mu _{k,t}(M)\big )^\frac{1}{2}\bigg ( \bigg (\int |{\mathbf {H}}|^2\, d\mu ^k_{t}\bigg )^\frac{1}{2} - (16\pi + 4 \kappa \mu ^k_{t} (M) )^{-\frac{1}{2}} \int |{\mathbf {H}}|^{2}\, d\mu ^k_{t}\bigg )\, dt \rightarrow 0 \end{aligned}$$

where \((\mu ^k_{t})_{t\ge 0}\) is the constructed Brakke flow starting at \(\Sigma _k\). We aim to let \(k \rightarrow \infty \) and construct a non-vanishing Brakke flow starting at \(\Sigma \).

By Lemma 2.5, there exists \(\delta >0\) and \(\eta >0\) such that

$$\begin{aligned} \mu ^k_{t}(M) \ge \eta \end{aligned}$$
(5.12)

for all \(0\le t < \delta \) and all k sufficiently large.

We can thus consider a subsequential limit as \(k \rightarrow \infty \) and obtain a limiting Brakke flow \(\{\mu _t\}_{t\ge 0}\) which satisfies (5.12) as well. Similarly as in the proof of Lemma 5.1 we obtain

$$\begin{aligned} \int _0^\delta \big (\mu _t(M)\big )^\frac{1}{2}\bigg ( \bigg (\int |{\mathbf {H}}|^2\, d\mu _{t}\bigg )^\frac{1}{2} -(16\pi + 4 \kappa \mu _{t} (M) )^{-\frac{1}{2}} \int |{\mathbf {H}}|^{2}\, d\mu _{t}\bigg )\, dt = 0 . \end{aligned}$$

Thus for a.e. \(t \in (0,\delta )\), using (5.12), we have

$$\begin{aligned} \int |{\mathbf {H}}|^2\, d\mu _{t} = 16 \pi + 4\kappa \mu _t(M) . \end{aligned}$$
(5.13)

Thus by Theorem 1.4 for a.e. \(t \in (0,\delta )\), \(\mu _t\) is the Radon measure associated to a smooth embedded 2-sphere \(\Sigma _t\) with density one, where the mean curvature vector has constant length and \(\Sigma _t\) is totally umbilic. Furthermore, it bounds a totally geodesic embedded 3-ball \(S_t\), with the mean curvature vector of \(\Sigma _t\) proportional to the unit conormal of \(S_t\) at every point in \(\Sigma _t\). \(S_t\) is isometric to a geodesic ball in the 3-dimensional model space such that the mean curvature of the boundary coincides with the one of \(\Sigma _t\). Since all the flows \(\{\mu ^k_t\}_{t\ge 0}\) are unit regular, see [SW16, § 4], White’s local regularity theorem [Whi05], implies that the convergence is smooth for \(0<t<\delta \) and the limiting flow \(\{\mu _t\}_{0<t<\delta }\) is smooth [and thus the above characterisation of \(\mu _t\) holds for all \(t \in (0,\delta )\)].

The smooth convergence implies that \(\mu ^k_t = \mu _{T^k_t}\) for any \(0<t<\delta \) and k sufficiently large. Recall that in the flat norm

$$\begin{aligned} \text {dist}(\Sigma _k,T^k_t) \le (t + t^{1/2}) {\mathbf {M}}[\Sigma _k] \end{aligned}$$

and \(\Sigma _k \rightarrow \Sigma \) in flat norm. This yields that

$$\begin{aligned} \Sigma _t \rightarrow \Sigma \end{aligned}$$

in flat norm. Since \(\Sigma _t\) converges smoothly to a limit as \(t \searrow 0\) this yields the claimed statement about \(\Sigma \).

In the case that \(\Sigma \) is a 2-dimensional flat chain mod 2, we work with flat chains mod 2 instead of integral currents. Note that Ilmanen’s elliptic regularisation scheme works analogously in this setting. All the other parts of the argument also directly carry over. The only point to note is that the interior regularity of Almgren [Alm00] has to be replaced by the corresponding result for flat chains mod 2 due to Federer [Fed70].\(\square \)

6 An Optimal Lower Bound on the Willmore Energy

In this section we give the proof of the optimal lower bound on the Willmore energy.

Proof of Theorem 1.4

We consider the vectorfield X given by

$$\begin{aligned} X:= \varphi (r) {\bar{\nabla }} r , \end{aligned}$$

where \(r(p):=\text {dist}_{M}(p,p_0)\) for a fixed \(p_0\in N\) and \(\varphi \in C^{0,1}_\text {loc}[0,\infty ), \varphi \ge 0\). Here \({\bar{\nabla }}\) denotes the gradient operator on M. The distance function to a point on such a manifold is smooth away from \(p_0\) and satisfies, see for example [Pet98]:

$$\begin{aligned} \begin{aligned} {\bar{\nabla }}r \ne&\ 0, \\ \text {Hess}(r)={\bar{\nabla }}^2r\ge&\ \Psi (r)\big (\text {id}-{\bar{\nabla }}r\otimes {\bar{\nabla }}r\big ) , \end{aligned} \end{aligned}$$

for \(p\ne p_0\), where \(\Psi (r)=1/r\) for \(\kappa = 0\) and \(\Psi (r)=\sqrt{\kappa }\cosh (\sqrt{\kappa }r)/\sinh (\sqrt{\kappa }r)\) for \(\kappa >0\) and the second inequality holds w.r.t. an orthonormal basis of \(T_pM\). For a point \(p\in \Sigma \), such that the tangent space of \(\Sigma \) exists at p we compute

$$\begin{aligned} \begin{aligned} \text {div}_{\Sigma }(X) =&\ \text {div}_\Sigma (\varphi {\bar{\nabla }}r) = \varphi \,\text {div}_\Sigma ({\bar{\nabla }}r) + \varphi ' {\bar{g}}(\nabla _\Sigma r, {\bar{\nabla }}r)\\ =&\ \varphi \, \text {tr}_{T_p\Sigma }\big (\text {Hess}(r)\big ) + \varphi '\big (1 - |({\bar{\nabla }} r)^\perp |^2\big )\\ \ge&\ \varphi \,\Psi \, \text {tr}_{T_p\Sigma }\big (\text {id}-{\bar{\nabla }}r\otimes {\bar{\nabla }}r\big ) + \varphi '\big (1 - |({\bar{\nabla }} r)^\perp |^2\big )\\ =&\ \varphi \,\Psi \big (1+ |({\bar{\nabla }} r)^\perp |^2\big ) + \varphi '\big (1 - |({\bar{\nabla }} r)^\perp |^2\big )\\ =&\ \varphi \, \Psi + \varphi ' + \big (\varphi \, \Psi - \varphi ') |({\bar{\nabla }} r)^\perp |^2 . \end{aligned} \end{aligned}$$
(6.1)

Pick any \(p_0\in \Sigma \) such that the density \(\Theta (p_0)\) exists and \(\Theta (p_0)\ge 1\).

The case\(\kappa =0\): Given \(\sigma >0\) we choose

$$\begin{aligned} \varphi (r) = \frac{r}{r_\sigma ^2} \end{aligned}$$

where \(r_\sigma =\max (r,\sigma )\). This gives

$$\begin{aligned} \int \text {div}_{\Sigma }(X)\, d\mu \ge 2\sigma ^{-2} \mu (B_\sigma (p_0)) + \int _{M \backslash B_\sigma (p_0)} 2 |X^\perp |^2\, d\mu . \end{aligned}$$
(6.2)

On the other hand, applying the divergence theorem yields

$$\begin{aligned} \int \text {div}_{\Sigma }(X)\, d\mu = - \sigma ^{-2} \int _{B_\sigma (p_0)} r\, {\bar{g}}({\bar{\nabla }}r, \mathbf {H})\, d\mu - \int _{ M \backslash B_\sigma (p_0)} {\bar{g}}(X, \mathbf {H})\, d\mu . \end{aligned}$$
(6.3)

Combining both equations yields

$$\begin{aligned} \begin{aligned} 2\sigma ^{-2}\mu (B_\sigma (p_0))+ 2\int _{M\backslash B_\sigma (p_0)} \bigg |\frac{1}{4}{\mathbf {H}}+ X^\perp \bigg |^2 d\mu \le \&\frac{1}{8}\int _{M\backslash B_\sigma (p_0)}|{\mathbf {H}}|^2\, d\mu \\&-\sigma ^{-2}\int _{B_\sigma (p_0)}r {\bar{g}}({\bar{\nabla }}r,{\mathbf {H}})\, d\mu . \end{aligned} \end{aligned}$$

Since

$$\begin{aligned} \lim _{\sigma \rightarrow 0}\sigma ^{-2}\mu (B_\sigma )(p_0) \ge \pi , \end{aligned}$$
(6.4)

we can take the limit \(\sigma \rightarrow 0\) to obtain

$$\begin{aligned} 2\pi + 2\int \bigg |\frac{1}{4}{\mathbf {H}}(x)+ \frac{\big ({\bar{\nabla }} r_{p_0}(x)\big )^\perp }{r_{p_0}(x)}\bigg |^2 d\mu (x) \le \frac{1}{8}\int |{\mathbf {H}}|^2\, d\mu \end{aligned}$$
(6.5)

for any \(p_0\) such that (6.4) holds.

The case\(\kappa >0\): we can assume w.l.o.g. via rescaling that \(\kappa =1\). We choose

$$\begin{aligned} \varphi _\sigma (r) = \frac{\sinh (r)}{(2\cosh (r) -2)_{\sigma ^2}} \end{aligned}$$

where \((2\cosh (r) -2)_{\sigma ^2} = \max (2\cosh (r) -2, \sigma ^2)\). The choice of \(\varphi _\sigma \) will become clear in the discussion of the equality case. We further denote \(\sigma '= \sigma '(\sigma )\) to be the solution of

$$\begin{aligned} 2\cosh (\sigma ') -2 = \sigma ^2. \end{aligned}$$

Note that,

$$\begin{aligned} \lim _{\sigma \rightarrow 0} \frac{\sigma '}{\sigma } = 1. \end{aligned}$$
(6.6)

Using \(\psi (r)= \cosh (r)/\sinh (r)\), for \(r<\sigma \) we have, suppressing the index \(\sigma \),

$$\begin{aligned} \varphi \,\psi + \varphi ' = 2 \sigma ^{-2}\cosh (r) \ge 2 \sigma ^{-2} \quad \text {and} \quad \varphi \, \psi - \varphi ' = 0 , \end{aligned}$$

and for \(r\ge \sigma \), noting that

$$\begin{aligned} \varphi '(r) = -\frac{1}{2\cosh (r) -2} \end{aligned}$$

we obtain

$$\begin{aligned} \varphi \, \psi + \varphi ' = \frac{1}{2}\quad \text {and} \quad \varphi \, \psi - \varphi ' = 2\varphi ^2. \end{aligned}$$

Inserting this into (6.1) gives

$$\begin{aligned} \int \text {div}_{\Sigma }(X)\, d\mu \ge 2\sigma ^{-2} \mu (B_{\sigma '}(p_0)) + \int _{M \backslash B_{\sigma '}(p_0)} \frac{1}{2}+2 |X^\perp |^2\, d\mu . \end{aligned}$$

Arguing as before, using (6.6), we arrive at

$$\begin{aligned} 2\pi + 2\int \bigg |\frac{1}{4}{\mathbf {H}}(x)+ \varphi _0(r_{p_0}(x)) \big ({\bar{\nabla }} r_{p_0}(x)\big )^\perp \bigg |^2 d\mu (x) + \int \frac{\kappa }{2}\, d\mu \le \frac{1}{8} \int |{\mathbf {H}}|^2\, d\mu \end{aligned}$$
(6.7)

for any \(p_0\) such that (6.4) holds.

The equality case for\(\kappa =0\): To see that \(\Sigma \) is a smoothly embedded 2-sphere with unit density, we can nearly verbatim follow the argument in [LS14, Proposition 2.1]. We include it for completeness. We first note that by equality in (6.5), since \(p_0 \in \mathrm{spt}\,\mu \) is arbitrary, we have that \(\Sigma \) has unit multiplicity:

$$\begin{aligned} \theta ^2(\mu )=1\quad \text {on } \mathrm{spt}\,\mu . \end{aligned}$$
(6.8)

Furthermore (6.5) gives that

$$\begin{aligned} {\mathbf {H}}(y)+ 4 \frac{\big ({\bar{\nabla }} r_{x}(y)\big )^{\perp _y}}{r_{x}(y)} \quad \text {for } \mu \text {-almost all } y\in \mathrm{spt}\,\mu , \end{aligned}$$

where \({}^{\perp _y}\) denotes the orthogonal projection onto \(T^\perp _y\mu \). In particular

$$\begin{aligned} {\mathbf {H}}(y) \perp T_y\mu \quad \text {for } \mu \text {-almost all } y\in \mathrm{spt}\,\mu . \end{aligned}$$
(6.9)

By Fubini’s Theorem, for \(\mu \)-almost all y it holds that

$$\begin{aligned} {\mathbf {H}}(y)+ 4 \frac{\big ({\bar{\nabla }} r_{x}(y)\big )^{\perp _y}}{r_{x}(y)} \quad \text {for } \mu \text {-almost all } x\in \mathrm{spt}\,\mu . \end{aligned}$$

We choose any \(y \in \mathrm{spt}\,\mu \) such that \(T_y\mu \) exists. If \({\mathbf {H}}(y) = 0\), then \(\mathrm{spt}\,\mu \subset \exp _y(T_y\mu )\). As in [LS14, Proposition 2.1] this contradicts the compactness of \(\mathrm{spt}\,\mu \). Hence \({\mathbf {H}}(y) \ne 0\) and we may assume that \({\mathbf {H}}(y) \perp T_y\mu \) by (6.9). By scaling and choosing exponential coordinates \(x = \exp _y\), we may assume that \(T_0\mu = \text {span}\{e_1,e_2\}, T\perp _0\mu = \text {span}\{e_3, \ldots ,e_n\}, {\mathbf {H}}(0)= 2e_3\) and we write \(\perp \) for the projection to \(\text {span}\{e_3, \ldots ,e_n\}\) in these coordinates. Denoting with \(\langle \cdot , \cdot \rangle \) the metric on \(T_yM\), we firstly get from the above for \(j=4, \ldots , n\), that

$$\begin{aligned} \begin{aligned} 0&= \left\langle {\mathbf {H}}(0), e_j \right\rangle = -4 \left\langle \frac{- x^\perp }{|x|^2}, e_j \right\rangle \\&= 4 \left\langle \frac{x^\perp }{|x|^2}, e_j \right\rangle = 4\frac{x_j}{|x|^2} \quad \text {for } \mu \text {-almost all } x\ne 0 \in \exp ^{-1}_y(\mathrm{spt}\,\mu ). \end{aligned} \end{aligned}$$

Thus \(\exp ^{-1}_y(\mathrm{spt}\,\mu ) \subset \text {span}\{e_1,e_2,e_3\}\). For \(j=3\) we obtain

$$\begin{aligned} 2 = \left\langle {\mathbf {H}}(0), e_3 \right\rangle = -4 \left\langle \frac{- x^\perp }{|x|^2}, e_3 \right\rangle = 4\frac{x_3}{|x|^2} \quad \text {for } \mu \text {-almost all } x\ne 0 \in \exp ^{-1}_y(\mathrm{spt}\,\mu ) . \end{aligned}$$

Thus \(2x_3 = |x|^2\) and again as in [LS14, Proposition 2.1] one sees that

$$\begin{aligned} \mu = {\mathcal {H}}^2 \,\textsf {L}\,\Sigma \end{aligned}$$

where

$$\begin{aligned} \Sigma = \exp _y\big (\partial B_1(e_3) \cap \text {span}\{e_1,e_2, e_3\}\big ). \end{aligned}$$

To construct the spanning flat 3-ball we argue as follows. Note first that we can repeat the same argument for every point \(y \in \Sigma \). Pick \(y_0 \in \Sigma \) such that

$$\begin{aligned} |{\mathbf {H}}(y_0)| = \max _{\Sigma } |{\mathbf {H}}| \end{aligned}$$
(6.10)

and denote \(r_0 = 2/|{\mathbf {H}}(y_0)|\). Applying the above argument at \(y_0\), but without scaling, we obtain

$$\begin{aligned} \Sigma = \exp _y\big (\partial B_{r_0}(r_0 e_3) \cap \text {span}\{e_1,e_2, e_3\}\big ) . \end{aligned}$$

We define

$$\begin{aligned} S = \exp _y\big ( {\overline{B}}_{r_0}(r_0 e_3) \cap \text {span}\{e_1,e_2, e_3\}\big ) . \end{aligned}$$

Claim

S with its induced metric \({\tilde{g}}\) is isometric via the exponential map at \(y_0\) to \(B_{r_0}(r_0 e_3) \subset \mathbb {R}^3\).

Following the proof of (6.5) we see that we have equality in (6.1) with \(\varphi = 1/r\) for every point \(x\ne y_0 \in \Sigma \). Since all geodesics connecting \(y_0\) with other points in \(x \in \Sigma \) intersect \(\Sigma \) at x non-tangentially, we have that the ambient sectional curvatures

$$\begin{aligned} \text {sec}_g({\bar{\nabla }}r_{y_0} \wedge V) = 0 , \end{aligned}$$

where V is any unit vector tangent to \(S_r:=\partial B_r(y_0) \cap S\) for \(0<r<2r_0\). The same argument gives that the principal curvatures along S of \(\partial B_r(y_0)\) are equal 1/r for \(0<r<2r_0\) and that intrinsically \(S_r\) is isometric via the exponential map at \(y_0\) to \(\partial B_r(0) \cap B_{r_0}(r_0 e_3) \subset \mathbb {R}^3\), written in polar coordinates around \(0 \in \mathbb {R}^3\). But the Gauss equations then also show that

$$\begin{aligned} \text {sec}_{{\tilde{g}}}(V \wedge W) = 0 \end{aligned}$$

for any two unit vectors V, W tangent to \(S_r\) for \(0<r<2r_0\). This proves the claim.

Note that this implies that the mean curvature vector \({\mathbf {H}}^S(x)\) of \(\Sigma \subset S\), seen as a submanifold of S has length \(2/r_0\) for all \(y \in \Sigma \). Since

$$\begin{aligned} {\mathbf {H}}^S(x) = \pi _{T_xS}\big ({\mathbf {H}}(x)\big ) \end{aligned}$$

the choice of \(y_0\) in (6.10) implies that

$$\begin{aligned} {\mathbf {H}}^S(x) = {\mathbf {H}}(x) \quad \forall \, x \in \Sigma . \end{aligned}$$

It remains to show S is totally geodesic. Pick any point \(x_0\in \Sigma \). By the argument before we have that

$$\begin{aligned} S = \exp _{x_0}\big ( B_{r_0}(r_0 e_3) \cap \text {span}\{e_1,e_2, e_3\}\big ) , \end{aligned}$$

where we have chosen \(e_1,e_2, e_3\) as before. But this implies that any extrinsic geodesic connecting \(x_0\) with \(x \ne x_0 \in \Sigma \) has the same length as the intrinsic geodesic in S connecting both points, and thus they both have to coincide. This shows that S is totally geodesic, which also implies that \(\Sigma \) is totally umbilic in M.

The equality case for\(\kappa >0\): We can again by scaling assume that \(\kappa =1\). The argument is completely analogous to the case \(\kappa =0\), the only thing to note is that the equation

$$\begin{aligned} 2 = 4 \, \frac{\sinh (|x|)}{2\cosh (|x|) -2}\, \frac{x_3}{|x|} \end{aligned}$$

describes the boundary of a geodesic sphere with mean curvature 2 in normal coordinates around the south pole in the 3-dimensional model space. \(\square \)