arXiv research

We consider smooth radial solutions to the Hamiltonian stationary equation which are defined away from the origin. We show that in dimension two all radial solutions on unbounded domains must be special Lagrangian. In contrast, for all higher dimensions there exist non-special Lagrangian radial solutions over unbounded…

We show that a smooth radially symmetric solution $u$ to the graphic Willmore surface equation is either a constant or the defining function of a half sphere in ${\mathbb R}^3$. In particular, radially symmetric entire Willmore graphs in ${\mathbb R}^3$ must be flat. When $u$ is a smooth radial solution over a puncture…

Study finds infinitely many non-radial solutions for negative scalar curvature in higher dimensions.

The paper examines solutions to a specific type of nonlinear equation in a disk, proving existence and uniqueness.

Study on radial solutions of Lane-Emden system on Cartan-Hadamard manifolds.

Study proves radial symmetry of solutions to certain nonlinear equations in space forms.

Study global geometry of toric nearly Kähler manifolds using multi-moment maps.

Study differential operators on non-compact harmonic manifolds, finding conditions for radial fundamental solutions and dense heat-semigroups.

We consider radial solutions to the fast diffusion equation $u_t=Δu^m$ on the hyperbolic space $\mathbb{H}^{N}$ for $N \ge 2$, $m\in(m_s,1)$, $m_s=\frac{N-2}{N+2}$. By radial we mean solutions depending only on the geodesic distance $r$ from a given point $o \in \mathbb{H}^N$. We investigate their fine asymptotics near…

We study conformally flat surfaces with prescribed Gaussian curvature, described by solutions $u$ of the PDE: $Δu(x)+K(x)\exp(2u(x))=0$, with $K(x)$ the Gauss curvature function at $x\in\RR^2$. We assume that the integral curvature is finite. For radially symmetric $K$ we introduce the notion of a least integrally curv…

Study finds unique radial solutions on manifolds using differential geometry and analysis.

Under the quadratic-decay-conditions of the radial curvatures of an end, we shall derive growth estimates of solutions to the eigenvalue equation and show the absence of eigenvalues.

Study proves radial symmetry in convex cones using subharmonic functions.

We study existence, uniqueness and stability of radial solutions of the Lane-Emden-Fowler equation $-Δ_g u=|u|^{p-1}u$ in a class of Riemannian models $(M,g)$ of dimension $n\ge 3$ which includes the classical hyperbolic space $\mathbb H^n$ as well as manifolds with sectional curvatures unbounded below. Sign properties…

The paper studies minimal resistance dynamics in radial fields, finding unique solutions for incompressible flows.

$σ_k$-Yamabe equations are conformally invariant equations generalizing the classical Yamabe equation. In an earlier work YanYan Li proved that an admissible solution with an isolated singularity at $0\in \mathbb R^n$ to the $σ_k$-Yamabe equation is asymptotically radially symmetric. In this work we prove that an admis…

Let $Ł$ be the Laplace operator on $\R ^d$, $d\geq 3$ or the Laplace Beltrami operator on the harmonic $NA$ group (in particular on a rank one noncompact symmetric space). For the equation $Łu - \varphi(\cdot,u)=0$ we give necessary and sufficient conditions for the existence of entire bounded or large solutions under…

We consider the asymptotic behaviour of positive solutions u of the conformal scalar curvature equation, Δu + n(n-2)/4 u^{(n+2)(n-2) = 0, in the neighbourhood of isolated singularities in the standard Euclidean ball. Although asymptotic radial symmetry for such solutions was proved some time ago, by Caffarelli, Gidas a…

Paper proves minimal resistance for a body in a fluid with decreasing density.

Classifies singular solutions to Liouville equation with constant Q-curvature metrics.

We determine the optimal structure of couplings for the \emph{Martingale transport problem} between radially symmetric initial and terminal laws $μ, ν$ on $\R^d$ and show the uniqueness of optimizer. Here optimality means that such solutions will minimize the functional $\E |X-Y|^p$ where $0<p \leq 1$, and the dimensio…

In 1996, Kirk Lancaster and David Siegel investigated the existence and behavior of radial limits at a corner of the boundary of the domain of solutions of capillary and other prescribed mean curvature problems with contact angle boundary data. In Theorem 3, they provide an example of a capillary surface in a unit disk…

Radial Basis Functions Neural Networks (RBFNNs) are tools widely used in regression problems. One of their principal drawbacks is that the formulation corresponding to the training with the supervision of both the centers and the weights is a highly non-convex optimization problem, which leads to some fundamentally dif…

Classifies solutions to critical sixth order equations with a singularity.

The study of the $k$-th elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so called $σ_k$ curvature, has produced many fruitful results in conformal geometry in recent years. In these studies, the deforming conformal factor is considered to be a solution of a fully nonlinea…

In this paper we extend a recent result of Collin-Rosenberg ({\it a solution to the minimal surface equation in the Euclidean disc has radial limits almost everywhere}) to a large class of differential operators in Divergence form. Moreover, we construct an example (in the spirit of \cite{CR2}) of a minimal graph in $\…

Long time existence and convergence to a circle is proved for radial graph solutions to a mean curvature type curve flow in warped product surfaces (under a weak assumption on the warp potential of the surface). This curvature flow preserves the area enclosed by the evolving curve, and this fact is used to prove a gene…

New Harnack inequality for curve shortening flow without convexity.

Soft-Radial Projection solves gradient saturation in constrained deep learning.

The paper classifies Cartan-Hadamard manifolds supporting optimal Sobolev inequalities.

Existence and uniqueness in ${\Bbb R}^{n,1}$ of entire spacelike hypersurfaces contained in the future of the origin $O$ and asymptotic to the light-cone, with scalar curvature prescribed at their generic point $M$ as a negative function of the unit vector $\overrightarrow{Om}$ pointing in the direction of $\overrighta…

We study uniqueness of positive solutions to the conformal scalar curvature equation on complete Riemannian manifolds with constant negative scalar curvature. We apply the results to show that conformal transformations on certain complete Riemannian manifolds of constant negative scalar curvature are isometries. We als…

The paper constructs many ancient solutions to the Yamabe flow on spheres.

Study on solutions to conformally invariant fourth order equations, classifying their properties.

Solves Christoffel-Minkowski problem and Hessian equations with radial symmetry.

This paper is devoted to the study of the singularity phenomenon of timelike extremal hypersurfaces in Minkowski spacetime $\mathbb{R}^{1+3}$. We find that there are two explicit lightlike self-similar solutions to a graph representation of timelike extremal hypersurfaces in Minkowski spacetime $\mathbb{R}^{1+3}$, the …

We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions.

We establish two comparison results between the solutions of a class of mean curvature equations and pieces of arcs of circles that satisfy the same Neumann boundary condition. Finally we present a number of examples where our estimates can be applied, some of them have a physical motivation.

We consider nonnegative solutions of the porous medium equation (PME) on a Cartan-Hadamard manifold whose negative curvature can be unbounded. We take compactly supported initial data because we are also interested in free boundaries. We classify the geometrical cases we study into quasi-hyperbolic, quasi-Euclidean and…

This paper develops efficient surrogate models for optimization of complex dynamical systems.

For any $n\ge 3$, $0<m\le (n-2)/n$, and constants $η>0$, $β>0$, $α$, satisfying $α\leβ(n-2)/m$, we prove the existence of radially symmetric solution of $\frac{n-1}{m}Δv^m+αv +βx\cdot\nabla v=0$, $v>0$, in $\R^n$, $v(0)=η$, without using the phase plane method. When $0<m<(n-2)/n$, $n\ge 3$, and $α=2β/(1-m)>0$, we prove…

Smooth even solutions found for a generalized convex geometry problem.

New exponential decay estimate for Hermitian Yang-Mills metrics near branch points.

We consider a fourth order partial differential equation in n-dimensional space introduced by Abreu in the context of Kähler metrics on toric orbifolds. Similarity solutions depending only on the radial coordinate in R^n are determined in terms of a second order ordinary differential equation. A local asymptotic analys…

The paper generalizes radial curvature bounds on manifolds.

Study examines maximal domains of radial harmonic functions across different curvature types.

We will give a new proof of a recent result of P.~Daskalopoulos, G.Huisken and J.R.King ([DH] and reference [7] of [DH]) on the existence of self-similar solution of the inverse mean curvature flow which is the graph of a radially symmetric solution in $\mathbb{R}^n$, $n\ge 2$, of the form $u(x,t)=e^{λt}f(e^{-λt} x)$ f…

Convexity properties are preserved under radial transformations in hyperbolic and spherical geometries.