The Difference of Topology Due to a False Critical Point at Infinity of the Third Kind

Abstract

As we pointed out in [2] and earlier, the variational problemJon C_βis not Fredholm because the functionalJonly controls the a component of $\dot x = a\xi + b\upsilon$(the director to the curve) and its gradient only controls $\smallint _0^tb$ in a weak sense (not even $\smallint _0^1|b|$). As we have shown in [2], this has several consequences: at any time t along a curve x(t) of C_β we can add progressively to x a piece of v-orbit which we describe as moving back and forth very rapidly. The new curve $\tilde x$ is not in Cß but it is easy to approximate it with curves of Cß; see [2], the introduction and pp. 234–244. Since α(v) is zero, J($\tilde x$) is equal toJ(x). We may repeat such a construction as many times as we please and thus we are led to replace the curve x by a more complex object which is made of x with several back and forth runs along ±v-orbits at various points of x. The flow of [2] does not increase the number of zeros of b and controls $\smallint _0^1|b|$ thus limiting, on decreasing deformation lines, the number of such fast runs along the v-orbits along x. However, they still take place and we have shown in [2] that they play a major role since, because of this phenomenon, some critical points or critical points at infinity may not induce any difference of topology in the level sets of the functional J; see [2], pp. 234—244. This occurs when, denoting cp_sthe one-parameter group of v, for some to and some so, (necessarily bounded away from zero) ${a_{x({t_0})}}(D{\varphi _{ - {s_0}}}(\xi )) - 1$ is positive. Considering then a piece of v-orbit through x(t₀), we can build the curve $\tilde x(t)$ obtained by joining to x a back and forth run along this piece of v-orbit. Opening it up as in [2], pp. 234–244, and adding a tiny ξ-piece at the rest point (so along the v-orbit) along the back and forth run, we can build a decreasing deformation (in C_β, after approximation) on a whole neighborhood of $\tilde x(t)$ i.e., a non-Fredholm decreasing deformation for a neighborhood of x. When xis a critical point or a critical point at infinity, no difference of topology in the level sets ofJwill then occur because x • $\tilde x(t)$ plays the role of another critical point or critical point at infinity cancellingx(hence of index equal to the index ofx + 1).