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 Cβis 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 Cβ 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 cpsthe 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 Cβ, 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). KeywordsInterior NodeDirac MassRest PointEdge SignCoalescence PhenomenonThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.