arXiv research

Stability of a new map derived from the equator map is analyzed.

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

Paper studies critical points of curvature energies in 4D.

Paper proves conjecture about Einstein metrics on manifolds with positive isotropic curvature.

The paper studies the smoothness of critical points of variational integrals on Hessian spaces.

Rigidity theorem for critical points of Allen-Cahn equation on S³.

Rigidity for 4D Willmore submanifolds with boundary.

A large class of semi-Hamiltonian systems of hydrodynamic type is interpreted as the equations governing families of critical points of functions obeying the classical linear Darboux equations for conjugate nets.The distinguished role of the Euler-Poisson-Darboux equations and associated Lauricella-type functions is em…

This paper deals with the existence of solutions to a class of fourth order nonlinear elliptic equations. The technique used relies on critical points theory. The solutions appeared as critical points of a functional restricted to a suitable manifold.In the case of constant coefficients we obtain the existence of tree …

Study finds multiple solutions for Gross-Pitaevskii equations on curved spaces.

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

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…

We show that the Green functions on flat tori can have either 3 or 5 critical points only. There does not seemto be any directmethod to attack this problem. Instead, we have to employ sophisticated non-linear partial differential equations to study it. We also study the distribution of number of critical points over th…

We solve the classifying problem raised by Fischer and Marsden for Bach flat static spaces. We also prove the conjecture about critical point equations proposed by Besse for Bach flat manifolds. Particularly in dimension 3, we derive an integral identity that allows us to obtain conformal flatness from the vanish of th…

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 …

We prove that the critical points of various energies such as the area, the Willmore energy, the frame energy for tori...etc among possibly branched immersions constrained to evolve within a smooth sub-manifold of the Teichmüller space satisfy the corresponding constrained Euler Lagrange equation. We deduce that critic…

Considering that the Seiberg-Witten functional satisfies the Palais-Smale Condition, up to gauge equivalence, the Minimax Principle can be applied on the moduli space to prove the existence of critical points, which correspond to solutions of the second-order SW-equations, up to gauge equivalence.

In this paper, we have studied the critical point equation (shortly, CPE) within the frame-work of Kenmotsu and almost Kenmotsu manifold satisfying certain nullity conditions. First, we prove that a complete Kenmotsu metric satisfies the CPE is Einstein and locally isometric to the hyperbolic space H2n+1. In case of Ke…

We study the problem of prescribing the Paneitz curvature on higher dimensional spheres. Particular attention is paid to the blow-up points, i.e. the critical points at infinity of the corresponding variational problem. Using topological tools and a careful analysis of the gradient flow lines in the neighborhood of suc…

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

Develops finite-gap solutions for Pohlmeyer--Lund--Regge equation and Lund--Regge curve evolution.

We study a functional, whose critical points couple Dirac-harmonic maps from surfaces with a two form. The critical points can be interpreted as coupling the prescribed mean curvature equation to spinor fields. On the other hand, this functional also arises as part of the supersymmetric sigma model in theoretical physi…

Proves critical points of ADM mass correspond to specific initial data sets.

The problem of prescribing conformally the scalar curvature of a closed Riemannian manifold as a given Morse function reduces to solving an elliptic partial differential equation with critical Sobolev exponent. Two ways of attacking this problem consist in subcritical approximations or negative pseudo gradient flows. W…

Study bounds the index of minimal submanifolds using energy measures and Yang-Mills-Higgs equations.

Extends Gauduchon's result to higher dimensions, showing balanced metrics.

Given a solution $u$ to a linear homogeneous second order elliptic equation with Lipschitz coefficients, we introduce techniques for giving improved estimates of the critical set $\Cr(u)\equiv \{x:|\nabla u|(x)=0\}$. The results are new even for harmonic functions on $\dR^n$. Given such a $u$, the standard {\it first o…

We study critical points of the Ginzburg-Landau (GL) functional and the abelian Yang-Mills-Higgs (YMH) functional on the sphere and the complex projective space, both equipped with the standard metrics. For the GL functional we prove that on $S^{n}$ with $n \geq 2$ and $\mathbb{CP}^{n}$ with $n \geq 1$, stable critical…

New mathematical surfaces without boundaries found.

In relativity, the energy of a moving particle depends on the observer, and the rest mass is the minimal energy seen among all observers. The Wang-Yau quasi-local mass for a surface in spacetime introduced in [7] and [8] is defined by minimizing quasi-local energy associated with admissible isometric embeddings of the …

Solves Besse conjecture on 3D manifolds, proving metric rigidity.

We prove that smooth critical points of the Möbius energy parametrized by arc-length are analytic. Together with the main result in \cite{BRS16} this implies that critical points of the Möbius energy with merely bounded energy are not only $C^\infty$ but also analytic. Our proof is based on Cauchy's method of majorants…

On a given closed connected manifold of dimension two, or greater, we consider the squared $L^2$-norm of the scalar curvature functional over the space of constant volume Riemannian metrics. We prove that its critical points have constant scalar curvature, and use this to show that a metric is a solution of the critica…

The relative equilibria of a symmetric Hamiltonian dynamical system are the critical points of the so-called augmented Hamiltonian. The underlying geometric structure of the system is used to decompose the critical point equations and construct a collection of implicitly defined functions and reduced equations describi…

The paper constructs solutions to a critical Dirac equation on spheres.

The paper compares inserting and stretching points for grid refinement near critical points.

Study examines wave equation decay and Strichartz estimates on conic manifolds.

The paper introduces a volume invariant for Hermitian-symplectic metrics and proves its critical points are Kähler.

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…

The paper finds solutions for the Yamabe equation on product manifolds.

Researchers examine global properties of a scalar curvature functional to solve the prescribed Ricci curvature problem.

Neural networks trained with actor-critic algorithms converge to ODEs under weak convergence analysis.

Paper finds relations between Willmore-type energies, weighted areas, and vertical potential energies for cylindrical critical points.

Study of critical points for 4D conformally invariant curvature energies.

It is shown that the Euler-Lagrange equations for a Lagrangian system on a Lie algebroid are obtained as the equations for the critical points of the action functional defined on a Banach manifold of curves. The theory of reduction and the relation with Lagrange multiplier method are also studied.

Variational reduction simplifies Lagrangian systems with scaling symmetries.

Study on cr-invariant variational problem for Legendrian curves in 3-sphere.

Functional for Spin(7) forms defined on compact manifolds.