arXiv research

Conformal invariance of two-dimensional variational problems is a condition known to enable a blow-up analysis of solutions and to deduce the removability of singularities. In this paper, we identify another condition that is not only sufficient, but also necessary for such a removability of singularities. This is the …

Study of harmonic maps from degenerating surfaces with free boundary.

The paper proves compactness of metrics on spheres with isolated singularities.

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

Paper analyzes solutions to equations on surfaces with boundary singularities.

Improved Beckner's inequality for axially symmetric functions on S^4.

The paper solves Serrin-type problems on Riemannian manifolds using new inequalities and identities.

The paper proves compactness of metrics with isolated singularities on a sphere.

Study Yang-Mills connections on four-manifolds, derive obstructions to bubbling.

Paper presents a new Pohozaev-Schoen identity for non-compact manifolds.

The classical Pohozaev identity constrains potential solutions of certain semilinear PDE boundary value problems. The Kazdan-Warner identity is a similar necessary condition important for the Nirenberg problem of conformally prescribing scalar curvature on the sphere. For dimensions $n\geq 3$ both identities are captur…

Warped product metrics are a class of Riemannian metrics on cross products $B \times F$ which have been well studied and provide a rich set of examples. In this paper we consider shrinking gradient Ricci solitons which are warped product metrics. We prove that if the curvature of the metric is bounded and the base $B$ …

Given two compact Riemannian manifolds with boundary $M_1$ and $M_2$ such that their respective boundaries $Σ_1$ and $Σ_2$ admit neighborhoods $Ω_1$ and $Ω_2$ which are isometric, we prove the existence of a constant $C$, which depends only on the geometry of $Ω_1\congΩ_2$, such that $|σ_k(M_1)-σ_k(M_2)|\leq C$ for eac…

Paper finds infinitely many solutions changing sign for critical fractional equations.

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…

The paper confirms Escobar's conjecture on Steklov eigenvalues.

We study asymptotic behavior of positive smooth solutions of the conformal scalar curvature equation in ${\bf R}^n$. We consider the case when the scalar curvature of the conformal metric is bounded between two positive numbers outside a compact set. It is shown that the solution has slow decay if the radial change is …

In this note we show how a generalized Pohozaev-Schoen identity due to Gover and Orsted \cite{GO} can be used to obtain some rigidity results for $V$-static manifolds and generalized solitons. We also obtain an Alexandrov type result for certain hypersurfaces in Einstein manifolds.

Study investigates singularity formation in $\alpha$-Yang-Mills-Higgs fields on spheres.

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

New cylindrical solutions found for Grushin-type problem.

We study some analytical and geometric properties of a two-dimensional nonlinear sigma model with gravitino which comes from supersymmetric string theory. When the action is critical w.r.t. variations of the various fields including the gravitino, there is a symmetric, traceless and divergence-free energy-momentum tens…

Study on Einstein deformations and desingularizations, finding some 4-orbifolds cannot be limits of smooth metrics.

We prove that in Riemannian manifolds the $k$-th Steklov eigenvalue on a domain and the square root of the $k$-th Laplacian eigenvalue on its boundary can be mutually controlled in terms of the maximum principal curvature of the boundary under sectional curvature conditions. As an application, we derive a Weyl-type upp…

This paper establishes certain existence and classification results for solutions to $SU(n)$ Toda systems with three singular sources at 0, 1, and $\infty$. First, we determine the necessary conditions for such an $SU(n)$ Toda system to be related to an $n$th order hypergeometric equation. Then, we construct solutions …

We prove a Pohozaev type identity for non-linear eigenvalue equations of the Dirac operator on Riemannian spin manifolds with boundary. As an application, we obtain that the mean curvature H of a conformal immersion S^{n}-> R^{n+1} satisfies $\int \partial_X H=0$ where X is a conformal vector field on S^{n} and where t…

Extends Toda system existence results to negative functions.

Study rigidity in Riemannian manifolds using Pohozoaev and P-function approaches.

Minimal harmonic maps proved for specific manifolds.

Brezis' open problem on harmonic maps resolved

Fermat constants fail to fully identify Clairaut constants for certain geodesics on a surface of revolution.

Study proves surfaces with constant curvature are simple shapes.

The paper classifies hypersurfaces in $\mathbb{H}^2 imes\mathbb{H}^2$ with constant curvature.

Study on biconservative hypersurfaces with constant scalar curvature in space forms.

Study constant angle surfaces in 4D Minkowski space, proving their properties.

We prove several facts about the Yamabe constant of Riemannian metrics on general noncompact manifolds and about S. Kim's closely related "Yamabe constant at infinity". In particular we show that the Yamabe constant depends continuously on the Riemannian metric with respect to the fine C^2-topology, and that the Yamabe…

Study CR Yamabe constant and CR structures on manifolds.

The paper studies curves of constant-ratio in pseudo-Galilean space.

The Cheeger constant increases under Ricci flow on spheres.

Study on higher-order Escobar constants for planar domains.

The paper defines new constants for $p$-Laplacian on manifolds.

Study classifies 3D self-shrinkers with constant second form norm.

In 1926 S. Nakajima (= A. Matsumura) showed that any convex body in $\R^3$ with constant width, constant brightness, and boundary of class $C^2$ is a ball. We show that the regularity assumption on the boundary is unnecessary, so that balls are the only convex bodies of constant width and brightness.

Study refines Siegel-Veech constants for abelian differentials.

Curves with constant torsion can be deformed arbitrarily.

New upper bound for Cheeger constant of hyperbolic surfaces.

Simplified proof for Cheeger's isoperimetric constant.

Study on isoparametric and constant-curvature hypersurfaces in Finsler spaces.