## Abstract

We first bound the codimension of an ancient mean curvature flow by the entropy. As a consequence, all blowups lie in a Euclidean subspace whose dimension is bounded by the entropy and dimension of the evolving submanifolds. This drastically reduces the complexity of the system. We use this in a major application of our new methods to give the first general bounds on generic singularities of surfaces in arbitrary codimension.

We also show sharp bounds for codimension in arguably some of the most important situations of general ancient flows. Namely, we prove that in any dimension and codimension any ancient flow that is cylindrical at \(-\infty \) must be a flow of hypersurfaces in a Euclidean subspace. This extends well-known classification results to higher codimension.

The bound on the codimension in terms of the entropy is a special case of sharp bounds for spectral counting functions for shrinkers and, more generally, ancient flows. Shrinkers are solutions that evolve by scaling and are the singularity models for the flow.

We show rigidity of cylinders as shrinkers in all dimension and all codimension in a very strong sense: Any shrinker, even in a large dimensional space, that is sufficiently close to a cylinder on a large enough, but compact, set is itself a cylinder. This is an important tool in the theory and is key for regularity; cf. (Colding and Minicozzi II in preprint, 2020).

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## References

- [AL]
U. Abresch and J. Langer, The normalized curve shortening flow and homothetic solutions,

*J. Differ. Geom.*,**23**(1986), 175–196. - [AHW]
B. Andrews, H. Li and Y. Wei, ℱ-stability for self-shrinking solutions to mean curvature flow,

*Asian J. Math.*,**18**(2014), 757–777. - [ADS]
S. B. Angenent, P. Daskalopoulos and N. Sesum,

*Uniqueness of two-convex closed ancient solutions to the mean curvature flow*, preprint. - [AS]
C. Arezzo and J. Sun, Self-shrinkers for the mean curvature flow in arbitrary codimension,

*Math. Z.*,**274**(2013), 993–1027. - [BW1]
J. Bernstein and L. Wang, A sharp lower bound for the entropy of closed hypersurfaces up to dimension six,

*Invent. Math.*,**206**(2016), 601–627. - [BW2]
J. Bernstein and L. Wang, Topology of closed hypersurfaces of small entropy,

*Geom. Topol.*,**22**(2018), 1109–1141. - [BW3]
J. Bernstein and L. Wang, A topological property of asymptotically conical self-shrinkers of small entropy,

*Duke Math. J.*,**166**(2017), 403–435. - [B]
S. Brendle, Embedded self-similar shrinkers of genus 0,

*Ann. Math. (2)*,**183**(2016), 715–728. - [BCh]
S. Brendle and K. Choi, Uniqueness of convex ancient solutions to mean curvature flow in \({\mathbf{R}}^{3}\),

*Invent. Math.*,**217**(2019), 35–76. - [Ca]
E. Calabi, Minimal immersions of surfaces in Euclidean spheres,

*J. Differ. Geom.*,**1**(1967), 111–125. - [Ca1]
M. Calle, Bounding dimension of ambient space by density for mean curvature flow,

*Math. Z.*,**252**(2006), 655–668. - [Ca2]
M. Calle,

*Mean curvature flow and minimal surfaces*, Thesis (Ph.D.), New York University, 2007. - [Cg]
S. Y. Cheng, Eigenfunctions and nodal sets,

*Comment. Math. Helv.*,**51**(1976), 43–55. - [CgLYa]
S. Y. Cheng, P. Li and S. T. Yau, Heat equations on minimal submanifolds and their applications,

*Am. J. Math.*,**106**(1984), 1033–1065. - [CxZh]
X. Cheng and D. Zhou, Eigenvalues of the drifted Laplacian on complete metric measure spaces,

*Commun. Contemp. Math.*,**19**, 1650001 (2017). - [ChHH]
K. Choi, R. Haslhofer and O. Hershkovits,

*Ancient low entropy flows, mean convex neighborhoods, and uniqueness*, preprint. - [ChHHW]
K. Choi, R. Haslhofer, O. Hershkovits and B. White,

*Ancient asymptotically cylindrical flows and applications*, preprint. - [ChY]
D. Christodoulou and S. T. Yau, Some remarks on the quasi-local mass, in

*Mathematics and General Relativity*, Contemp. Math., vol. 71,*Santa Cruz, CA*,*1986*, pp. 9–14, AMS, Providence, 1988. - [CIM]
T. H. Colding, T. Ilmanen and W. P. Minicozzi II, Rigidity of generic singularities of mean curvature flow,

*Publ. Math. IHÉS*,**121**(2015), 363–382. - [CIMW]
T. H. Colding, T. Ilmanen, W. P. Minicozzi II and B. White, The round sphere minimizes entropy among closed self-shrinkers,

*J. Differ. Geom.*,**95**(2013), 53–69. - [CM1]
T. H. Colding and W. P. Minicozzi II, Harmonic functions with polynomial growth,

*J. Differ. Geom.*,**46**(1997), 1–77. - [CM2]
T. H. Colding and W. P. Minicozzi II, Harmonic functions on manifolds,

*Ann. Math. (2)*,**146**(1997), 725–747. - [CM3]
T. H. Colding and W. P. Minicozzi II, Weyl type bounds for harmonic functions,

*Invent. Math.*,**131**(1998), 257–298. - [CM4]
T. H. Colding and W. P. Minicozzi II, Liouville theorems for harmonic sections and applications,

*Commun. Pure Appl. Math.*,**52**(1998), 113–138. - [CM5]
T. H. Colding and W. P. Minicozzi II,

*Minimal Surfaces*, Courant Lecture Notes in Mathematics, vol. 4, New York University, Courant Institute of Mathematical Sciences, New York, 1999. - [CM6]
T. H. Colding and W. P. Minicozzi II, Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman,

*J. Am. Math. Soc.*,**18**(2005), 561–569. - [CM7]
T. H. Colding and W. P. Minicozzi II,

*A Course in Minimal Surfaces*, Graduate Studies in Mathematics, vol. 121, AMS, Providence, 2011. - [CM8]
T. H. Colding and W. P. Minicozzi II, Generic mean curvature flow I; generic singularities,

*Ann. Math.*,**175**(2012), 755–833. - [CM9]
T. H. Colding and W. P. Minicozzi II, Uniqueness of blowups and Lojasiewicz inequalities,

*Ann. Math.*,**182**(2015), 221–285. - [CM10]
T. H. Colding and W. P. Minicozzi II, In search of stable geometric structures,

*Not. Am. Math. Soc.*,**66**(2019), 1785–1791. - [CM11]
T. H. Colding and W. P. Minicozzi II,

*Regularity of elliptic and parabolic systems*, preprint (2019). - [CM12]
T. H. Colding and W. P. Minicozzi II,

*Optimal bounds for ancient caloric functions*, preprint. - [CM13]
T. H. Colding and W. P. Minicozzi II, Liouville properties,

*ICCM Not.*,**7**(2019), 16–26. - [CM14]
T. H. Colding and W. P. Minicozzi II,

*Rigidity of singularities for Ricci flow*, preprint. - [CM15]
T. H. Colding and W. P. Minicozzi II, Uniqueness of blowups for Ricci flow, in preparation.

- [dCW]
M. do Carmo and N. Wallach, Minimal immersions of spheres into spheres,

*Ann. Math. (2)*,**93**(1971), 43–62. - [CtHi]
R. Courant and D. Hilbert,

*Methods of Mathematical Physics I*, Interscience, New York, 1955. - [E1]
K. Ecker,

*Regularity Theory for Mean Curvature Flow*, Progress in Nonlinear Differential Equations and Their Applications, vol. 57, Birkhäuser Boston, Boston, 2004. - [E2]
K. Ecker, Partial regularity at the first singular time for hypersurfaces evolving by mean curvature,

*Math. Ann.*,**356**(2013), 217–240. - [E3]
K. Ecker, Logarithmic Sobolev inequalities on submanifolds of Euclidean space,

*J. Reine Angew. Math.*,**522**(2000), 105–118. - [EI]
A. El Soufi and S. Ilias, Immersions minimales, premiére valeur propre du laplacien et volume conforme,

*Math. Ann.*,**275**(1986), 257–267. - [GKS]
Z. Gang, D. Knopf and I. M. Sigal,

*Neckpinch Dynamics for Asymmetric Surfaces Evolving by Mean Curvature Flow*, Memoirs of American Mathematical Society, vol. 253 (1210), 2018. - [GNY]
A. Grigor’yan, Y. Netrusov and S. T. Yau, Eigenvalues of elliptic operators and geometric applications, in

*Surveys in Differential Geometry, vol. IX*, Surv. Differ. Geom., vol. 9, pp. 147–217, IP, Somerville, 2004. - [Ham]
R. Hamilton, Formation of singularities in the Ricci flow, in

*Surveys in Differential Geometry, vol. II*,*Cambridge, MA*,*1993*, pp. 7–136, International Press, Cambridge, 1993. - [Ha]
J. Harris,

*Algebraic Geometry. A First Course*, Graduate Texts in Mathematics, vol. 133, Springer, New York, 1992. - [He]
J. Hersch, Quatre properiétés isopérimétriques de membranes sphériques homogénes,

*C. R. Acad. Sci. Paris*,**270**(1970), 1645–1648. - [Has]
R. Haslhofer, Uniqueness of the bowl soliton,

*Geom. Topol.*,**19**(2015), 2393–2406. - [HH]
R. Haslhofer and O. Hershkovits, Ancient solutions of the mean curvature flow,

*Commun. Anal. Geom.*,**24**(2016), 593–604. - [H]
O. Hershkovits, Translators asymptotic to cylinders,

*Crelle*, to appear. - [HW]
O. Hershkovits and B. White, Sharp entropy bounds for self-shrinkers in mean curvature flow,

*Geom. Topol.*,**23**(2019), 1611–1619. - [Hu]
G. Huisken, Asymptotic behavior for singularities of the mean curvature flow,

*J. Differ. Geom.*,**31**(1990), 285–299. - [I]
T. Ilmanen,

*Singularities of mean curvature flow of surfaces*, preprint (1995). - [Ka]
M. Karpukhin, On the Yang-Yau inequality for the first Laplace eigenvalue,

*Geom. Funct. Anal.*,**29**(2019), 1864–1885. - [KZ]
D. Ketover and X. Zhou, Entropy of closed surfaces and min-max theory,

*J. Differ. Geom.*,**110**(2018), 31–71. - [K]
N. Korevaar, Upper bounds for eigenvalues of conformal metrics,

*J. Differ. Geom.*,**37**(1993), 73–93. - [LL]
Y.-I. Lee and Y.-K. Lue, The stability of self-shrinkers of mean curvature flow in higher co-dimension,

*Trans. Am. Math. Soc.*,**367**(2015), 2411–2435. - [LZ]
F. H. Lin and Q. S. Zhang, On ancient solutions of the heat equation,

*Commun. Pure Appl. Math.*,**LXXII**(2019), 2006–2028. - [Wa]
M.-T. Wang, Lectures on mean curvature flows in higher codimensions, in

*Handbook of Geometric Analysis. No. 1*, Adv. Lect. Math. (ALM), vol. 7, pp. 525–543, Int. Press, Somerville, 2008. - [Mi]
M. Micallef, Stable minimal surfaces in Euclidean space,

*J. Differ. Geom.*,**19**(1984), 57–84. - [Mu]
J. Munkres,

*Topology*, 2nd ed., Prentice Hall, Upper Saddle River, 2000. - [W1]
B. White, A local regularity theorem for mean curvature flow,

*Ann. Math. (2)*,**161**(2005), 1487–1519. - [W2]
B. White, The size of the singular set in mean curvature flow of mean-convex sets,

*J. Am. Math. Soc.*,**13**(2000), 665–695. - [W3]
B. White, Partial regularity of mean-convex hypersurfaces flowing by mean curvature,

*Int. Math. Res. Not.*,**4**(1994), 185–192. - [YY]
P. C. Yang and S. T. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds,

*Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4)*,**7**(1980), 55–63. - [Z]
J. Zhu, On the entropy of closed hypersurfaces and singular self-shrinkers,

*J. Differ. Geom.*,**114**(2020), 551–593.

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The authors were partially supported by NSF Grants DMS 1812142 and DMS 1707270.

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Colding, T.H., Minicozzi, W.P. Complexity of parabolic systems.
*Publ.math.IHES* **132, **83–135 (2020). https://doi.org/10.1007/s10240-020-00117-x

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