arXiv research

Paper studies unique interior points and estimates for generalized translating soliton problems.

The paper studies harmonic maps between surfaces homotopic to a covering map, proving uniqueness and injectivity of Hopf differential.

Near a birth-death critical point in a one-parameter family of gradient flows, there are precisely two Morse critical points of index difference one on the birth side. This paper gives a self-contained proof of the folklore theorem that these two critical points are joined by a unique gradient trajectory up to time-shi…

The paper classifies and analyzes the stability of elastic curves with fixed endpoints.

On the space of positive 3-forms on a seven-manifold, we study a natural functional whose critical points induce metrics with holonomy contained in $G_2$. We prove short-time existence and uniqueness for its negative gradient flow. Furthermore, we show that the flow exists for all times and converges modulo diffeomorph…

Hard to approximate critical points for simple nonconvex functions.

Given an elliptic integrand of class $\mathscr{C}^{3}$, we prove that finite unions of disjoint open Wulff shapes with equal radii are the only volume-constrained critical points of the anisotropic surface energy among all sets with finite perimeter and reduced boundary almost equal to its closure.

The paper proves unique blow-down for critical points of a Yang-Mills-Higgs functional.

We introduce the gradient flow of the Seiberg-Witten functional on a compact, orientable Riemannian 4-manifold and show the global existence of a unique smooth solution to the flow. The flow converges uniquely in $C^\infty$ up to gauge to a critical point of the Seiberg-Witten functional.

We show that the energy density of critical points of a class of conformally invariant variational problems with small energy on the unit 2-disk B_1 lies in the local Hardy space h^1(B_1). As a corollary we obtain a new proof of the energy convexity and uniqueness result for weakly harmonic maps with small energy on B_…

New result on critical points of Bethe free energy under deformation retracts.

New spinorial functional connects Perelman's W- and F-functionals.

On the universal bundle of unit spinors we study a natural energy functional whose critical points, if dim M \geq 3, are precisely the pairs (g, φ) consisting of a Ricci-flat Riemannian metric g together with a parallel g-spinor φ. We investigate the basic properties of this functional and study its negative gradient f…

Critical hypersurfaces with boundary have unique shapes and properties.

Introduces a new $G_2$-Hilbert functional in $G_2$-geometry.

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…

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

Uniqueness of nondegenerate blowups for planar networks shown.

Study on a pinning model with random walk increments, showing convergence to a critical disordered pinning measure.

The study examines critical points of a moment map for associative algebras.

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

For a fixed smooth map $u_0$ between two Riemann surfaces $Σ$ and $S$ with non-zero degree, we consider the energy function on Teichmüller space $\mc{T}$ of $Σ$ that assigns to a complex structure $t\in \mc{T}$ on $Σ$ the energy of the harmonic map $u_t:Σ_t:=(Σ,t) \to S$ homotopic to $u_0$. We prove that the energy fun…

Proves uniqueness of catenoid-like shapes in a ball.

Proves mass-capacity inequalities for critical area-normalized capacitors, improving Schwarzschild metric uniqueness.

We define a quantisation of the J-flow over a projective complex manifold. As corollaries, we obtain new proofs of uniqueness of critical points of the J-flow and that these critical points achieve the absolute minimum of an associated energy functional. We show that the existence of a critical point of the J-flow impl…

The paper proves smooth convergence of evolving hypersurfaces to critical points.

Study on scalar curvature minimizability loss and saddle point solutions.

We consider the problem of mean-variance portfolio optimization for a generic covariance matrix subject to the budget constraint and the constraint for the expected return, with the application of the replica method borrowed from the statistical physics of disordered systems. We find that the replica symmetry of the so…

We prove the existence of a unique global weak solution to the full bosonic string heat flow from closed Riemannian surfaces to an arbitrary target under smallness conditions on the two-form and the scalar potential. The solution is smooth with the exception of finitely many singular points. Finally, we discuss the con…

The critical catenoid is uniquely determined by certain symmetries of its boundary.

In this paper, based on the local comparison principle in [12], we study the local behavior of the difference of two spacelike graphs in a neighborhood of a second contact point. Then we apply it to the constant mean curvature equation in 3-dimensional Lorentz-Minkowski space $\mathbb{L}^3$ and get the uniqueness of cr…

Derives scalar reduction for generalized Kähler-Ricci solitons, proving uniqueness.

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 …

This survey paper contains an elementary exposition of Casson and Rivin's technique for finding the hyperbolic metric on a 3-manifold M with toroidal boundary. We also survey a number of applications of this technique. The method involves subdividing M into ideal tetrahedra and solving a system of gluing equations to f…

Study Zoll manifolds with boundary, showing unique geodesic properties.

Infinity-harmonic functions linked to IMCF clusters, revealing new properties in 2D.

The paper challenges the assumption of a unique global time in financial markets, highlighting market incompleteness.

In the spirit of recent work of Lamm, Malchiodi and Micallef in the setting of harmonic maps, we identify Yang-Mills connections obtained by approximations with respect to the Yang-Mills α-energy. More specifically, we show that for the SU(2) Hopf fibration over the four sphere, for sufficiently small α values the SO(4…

The paper studies metrics that match prescribed geodesics and introduces a variational problem.

For a knot $K\subset S^3$, its exterior $E(K) = S^3\backslashη(K)$ has a singular foliation by Seifert surfaces of $K$ derived from a circle-valued Morse function $f\colon E(K)\to S^1$. When $f$ is self-indexing and has no critical points of index 0 or 3, the regular levels that separate the index-1 and index-2 critica…

This paper reverses a construction by merging boundary critical points into an interior one.

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

The paper explores unique continuation properties for polyharmonic maps between Riemannian manifolds.

We study a class of elastic energy functionals for maps between planar domains (among them the so-called squared distance functional) whose critical points (elastic maps) allow a far more complete theory than one would expect from general elasticity theory. For some of these functionals elastic maps even admit a "Weier…

The study of equidistants for families of surfaces, focusing on specific ratios of tangent planes.

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

In this paper, we consider the smooth map from a Riemannian manifold to the standard Euclidean space and the p-Ginzburg-Landau energy. Under suitable curvature conditions on the domain manifold, some Liouville type theorems are established by assuming either growth conditions of the p-Ginzburg-Landau energy or an asymp…

Minimal surfaces in spheres have unique energy properties.