arXiv research

The paper studies critical points in overparameterized neural networks, identifying a star locus and degenerate critical points.

We study critical points of holomorphic sections of $\ocal(m)$ on $\CP^n$. For quadrics, we give a complete discription of their critical points. When $n=1$, we prove a spherical Gauss-Lucas theorem. For general situation, we prove that a general section has all its critical points isolated and non-degenerate.

The study confirms a conjecture about critical points of smooth functions.

We prove that the level sets of a real C^s function of two variables near a non-degenerate critical point are of class C^[s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at hyperbolic points or elliptic points, and in particular at umbilic points. We al…

Paper proves convex domains have one maximum for semi-stable solutions.

A new invariant captures geometric features of circle embeddings.

Let C be a hyperelliptic Riemann surface. We show that the hyperelliptic Weierstrass points of C are non-degenerated critical points of Morse index +2 of the curvature function K of the Theta metric on C (called also Bergman metric). When the genus of C is two, we compute all critical points of K and we show in this ca…

In this note, we give a proof of the famous theorem of M. Morse dealing with the cancellation of a pair of non-degenerate critical points of a smooth function. Our proof consists of a reduction to the one-dimensional case where the question becomes easy to answer.

The paper proves stability of critical points for conformally invariant Lagrangians.

We show the existence of a local foliation of a three dimensional Riemannian manifold by critical points of the Willmore functional subject to a small area constraint around non-degenerate critical points of the scalar curvature. This adapts a method developed by Rugang Ye to construct foliations by surfaces of constan…

Study shows limits of volume-constrained sets are finite unions of Wulff shapes.

We establish regularity results for critical points to energies of immersed surfaces depending on the first and the second fundamental form exclusively. These results hold for a large class of intrinsic elliptic Lagrangians which are sub-critical or critical. They are derived using uniform $ε-$regularity estimates whic…

The minimal number of critical points is studied for smooth functions on closed manifolds.

The paper calculates critical configurations and Morse indices for polygons on circles or ellipses.

Rigidity of critical eigensections on spheres proven.

Smooth functions on manifolds with degenerate singular submanifolds

We study topology change in (2+1)D gravity coupling with non-Abelian SO(2,1) Higgs field from the point of view of Morse theory. It is shown that the Higgs potential can be identified as a Morse function. The critical points of the latter (i.e. loci of change of the spacetime topology) coincide with zeros of the Higgs …

Two non-Morse-Bott Chern-Simons functions on homology 3-spheres.

In this article we introduce and investigate a new two-parameter family of knot energies $TP^{(p,q)}$ that contains the tangent-point energies. These energies are obtained by decoupling the exponents in the numerator and denominator of the integrand in the original definition of the tangent-point energies. We will firs…

In this paper, the following three are shown. (1) For a $C^\infty$ convex integrand $γ: S^n\to \mathbb{R}_+$, its dual convex integrand $δ: S^n\to \mathbb{R}_+$ is of class $C^\infty$. (2) For a stable convex integrand $γ: S^n\to \mathbb{R}_+$, its dual convex integrand $δ: S^n\to \mathbb{R}_+$ is stable. (3) Let $γ: S…

This article is devoted to the variational study of two functions defined over some Teichmueller spaces of hyperbolic surfaces. One is the systole of geodesic loops based at some fixed point, and the other one is the systole of arcs.\par For each of them we determine all the critical points. It appears that the systole…

Let $(M,g)$ be a 3-dimensional Riemannian manifold. The goal of the paper it to show that if $P_{0}\in M$ is a non-degenerate critical point of the scalar curvature, then a neighborhood of $P_{0}$ is foliated by area-constrained Willmore spheres. Such a foliation is unique among foliations by area-constrained Willmore …

The paper proves that Gaussian field critical points have finite moments.

We give the details of the proof of the equality between the critical groups, with respect the H^1 and C^1 topology, at a non-degenerate critical point of the energy functional of a non-reversible Finsler manifold (M,F), defined on the Hilbert manifold of the H^1 curves connecting two given points on M.

In this paper, we study a three-dimensional Ricci-degenerate Riemannian manifold $(M^3,g)$ that admits a smooth nonzero solution $f$ to the equation \begin{align} \label{a1a} \nabla df=ψRc+φg, \end{align} where $ψ,φ$ are given smooth functions of $f$, $Rc$ is the Ricci tensor of $g$. Spaces of this type include various…

Non-degeneracy of critical points proven for manifold's squared norm of second fundamental form.

New formulae connect topological and geometric properties of singular spaces.

In the present work we establish a quantization result for the angular part of the energy of solu- tions to elliptic linear systems of Schrödinger type with antisymmetric potentials in two dimension. This quantization is a consequence of uniform Lorentz-Wente type estimates in degenerating annuli. We derive from this a…

In this paper, we investigate simultaneous properties of a convex integrand $γ$ and its dual $δ$. The main results are the following three. (1) For a $C^\infty$ convex integrand $γ: S^n\to \mathbb{R}_+$, its dual convex integrand $δ: S^n\to \mathbb{R}_+$ is of class $C^\infty$ if and only if $γ$ is a strictly convex in…

The Morse function $f$ near a non-degenerate critical point $p$ is understood topologically, in the light of Morse's lemma. However, Morse's lemma standardizes the function $f$ itself, providing little information of how the gradient $\nabla f$ behaves. In this paper, we prove an analytical analogue of Morse's lemma, s…

Maps of degree 1 and critical points on manifolds are studied.

A Morse function f on a manifold with corners M allows the characterization of the Morse data for a critical point by the Morse index. In fact, a modified gradient flow allows a proof of the Morse theorems in a manner similar to that of classical Morse theory. It follows that M is homotopy equivalent to a CW-complex wi…

Embedding principle explains loss landscape of deep neural networks.

In this note, we study the curvature flow to Nirenberg problem on $S^2$ with non-negative nonlinearity. This flow was introduced by Brendle and Struwe. Our result is that the Nirenberg problems has a solution provided the prescribed non-negative Gaussian curvature $f$ has its positive part, which possesses non-degenera…

The paper examines critical points of solutions to a surface equation in 3D spacelike spaces.

Proves existence of sphere foliations with prescribed mean curvature on Riemannian manifolds.

In this note, we study Q-curvature flow on $S^4$ with indefinite nonlinearity. Our result is that the prescribed Q-curvature problem on $S^4$ has a solution provided the prescribed Q-curvature $f$ has its positive part, which possesses non-degenerate critical points such that $Δ_{S^4} f\not=0$ at the saddle points and …

We consider a family of variational problems on a Hilbert manifold parameterized by an open subset of a Banach manifold, and we discuss the genericity of the nondegeneracy condition for the critical points. Based on an idea of B. White, we prove an abstract genericity result that employs the infinite dimensional Sard--…

Tight isoparametric hypersurfaces in spheres have minimal critical points.

Study on eigenfunctions on sphere configurations, proving non-existence and construction of critical eigensections.

We prove that the conformal immersions of complex two tori into $S^3$ which locally minimize their conformal volume in their conformal class all satisfy some elliptic PDE. We prove that they are either minimal tori, CMC flat tori, elliptic conformally constrained minimal tori or critical point of the area under some fi…

It is a consequence of the Morse-Bott Lemma on Banach spaces that a smooth Morse-Bott function on an open neighborhood of a critical point in a Banach space obeys a Lojasiewicz gradient inequality with the optimal exponent one half. In this article we prove converses for analytic functions on Banach spaces: If the Loja…

Paper develops estimates for Lagrangian phase changes in 2D.

We generalize the Novikov inequalities for 1-forms in two different directions: first, we allow non-isolated critical points (assuming that they are non-degenerate in the sense of R.Bott), and, secondly, we strengthen the inequalities by means of twisting by an arbitrary flat bundle. The proof uses Bismut's modificatio…

Let $γ$ be a non-degenerate Ustilovsky geodesic in $Ham (M, ω)$ generated by $H$. We give a simple proof of a generalization of the conjecture stated in \cite{virtmorse}, relating the Morse index of $γ$, as a critical point of the Hofer length functional, with the Conley Zehnder index of the extremizers of $H$, consid…

The study proves the finiteness of moments for Gaussian field zeros and critical points.

For a monotonically advancing front, the arrival time is the time when the front reaches a given point. We show that it is twice differentiable everywhere with uniformly bounded second derivative. It is smooth away from the critical points where the equation is degenerate. We also show that the critical set has finite …

By a Morse function on a compact manifold with boundary we mean a real-valued function without critical points near the boundary such that its critical points as well as the critical points of its restriction to the boundary are all non-degenerate. For such Morse functions, Saeki and Yamamoto have previously defined a …