Morse theory for analytic functions on surfaces

Abstract

In this paper we deal with analytic functions f : S → R defined on a compact two dimensional Riemannian surface S whose critical points are semi degenerated (critical points having a non identically vanishing Hessian). To any element p of the set of semi degenerated critical points Q we assign an unique index which can take the values −1, 0 or 1, and prove that Q is made up of finitely many (critical) points with non zero index and embedded circles. Further, we generalize the famous Morse result by showing that the sum of the indexes of the critical points of f equals χ(S), the Euler characteristic of S. As an intermediate result we locally describe the level set of f near a point p ∈ Q. We show that the level set f −1 (f (p)) is either a) the set {p}, or b) the graph of a smooth curve passing through p, or c) the graphs of two smooth curves tangent at p or d) the graphs of two smooth curves building at p a cusp shape. (2000): 37B30, 58E05, 30F30. Mathematics Subject Classification

A Morse Lemma for Degenerate Critical Points of Solutions of Nonlinear Equations in ℝ2

“…In our opinion (1.4) is the most relevant results of the Theorems (see [3] for some properties of solutions satisfying (1.17)). In particular we get that there is no u verifying (1.1) such that u(x, y) ∼ u(0, 0) − x 4 − y 4 in a neighborhood of (0, 0)!…”

Taller in the saddle: constraining CMB physics using saddle points

“…[1,26], where hot spots are identified as pixels in the map that have a greater value than their nearestneighbour pixels, and similarly for cold spots. We, however, choose to find the extrema in a manner that is consistent with our method of finding saddle points, so that, for example, we find an equal number of extrema and saddle points, consistent with Morse theory [25]. Using the nearest neighbour-method for identifying extrema returns 180,000 extrema, whereas the critical point method returns 274,000 extrema, which is equal to the number of saddle points found.…”

“…For example, one can naively compute the number of extrema and saddle points by conditioning on being at a critical point, and then compute the relative probabilities of that point being an extremum or a saddle point. This calculation gives the result that saddle points make up 1/ √ 3 ≈ 58% of all critical points, whereas a well-known theorem from Morse theory, and effectively going back to Euler and Poincaré, states that for analytic functions in the plane the numbers of saddle points and extrema are equal, and that they differ by exactly 2 points on the sphere [25].…”

“…We can see that the above conclusion is incorrect by comparing with a well-known result from Morse theory, which says that all functions on the sphere (subject to very mild hypotheses) have precisely two more extrema than saddle points. In general, the number of saddle points exceeds the number of extrema in an analytic function on a manifold by the Euler characteristic of the manifold [25]. Suppose that we have a Gaussian random process on the sphere, whose power spectrum contains mostly small-scale power, so that the flat-sky approximation is good.…”

Non-existence of geometric minimal foliations in hyperbolic three-manifolds

“…Secondly (4.11) shows that at non-zero critical points, we have that the Hessian Hess s does not vanish identically. Hence, any arc in a level subset of s is smooth (one may see [AP05] for more details). As noted in the proof of Lemma 4.2, if p ∈ C, then if p is a saddle point, then the s(p)-level set of s must contain a smooth arc, and hence s attains its global maximum at p. Since s ≥ 0 but has zeroes at only the (finitely many) zeroes of A (see Remark (2.3)), we see that the only critical values obtainable are either global maxima or zeroes (global minima): these account for all the critical points in C.…”