arXiv research

Study constant mean curvature surfaces with integrable boundary conditions.

Proves a theorem for normal distributions on manifolds with boundary.

We give time-slicing path integral formulas for solutions to the heat equation corresponding to a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold with boundary. More specifically, we show that such a solution can be approximated by integrals over finite-dimens…

Harmonic functions $u:{\mathbb R}^n \to {\mathbb R}^m$ are equivalent to integral manifolds of an exterior differential system with independence condition $(M,{\mathcal I},ω)$. To this system one associates the space of conservation laws ${\mathcal C}$. They provide necessary conditions for $g:{\mathbb S}^{n-1} \to M$ …

Study on smoothness of solutions to nonlinear equations on Riemannian manifolds.

New varifolds with capillary boundary properties studied.

The uniform boundary condition in a normed chain complex asks for a uniform linear bound on fillings of null-homologous cycles. For the $\ell^1$-norm on the singular chain complex, Matsumoto and Morita established a characterisation of the uniform boundary condition in terms of bounded cohomology. In particular, spaces…

Solves Einstein vacuum equations with specific boundary conditions.

Paper derives a Reilly type integral formula and applies it to inequalities and eigenvalue problems.

Let M be a compact manifold with boundary. In this paper, we discuss some rigidity theorems of metrics in a same conformal class that fixes the boundary and satisfy certain integral conditions on the the scalar curvatures and the mean curvatures on the boundary. No condition on the first eigenvalues of operators is nee…

Novel boundary integral equations for Dirac operators in 3D Lipschitz domains.

Quantitative Sobolev extensions lead to Neumann heat kernel bounds.

The Poisson equation on manifolds plays an fundamental role in many applications. Recently, we proposed a novel numerical method called the Point Integral method (PIM) to solve the Poisson equations on manifolds from point clouds. In this paper, we prove the convergence of the point integral method for solving the Pois…

We prove a Li-Yau gradient estimate for positive solutions to the heat equation, with Neumann boundary conditions, on a compact Riemannian submanifold with boundary ${\bf M}^n\subseteq {\bf N}^n$, satisfying the integral Ricci curvature assumption: \begin{equation} D^2 \sup_{x\in {\bf N}} \left( \oint_{B(x,D)} |Ric^-|^…

We prove that some Riemannian manifolds with boundary under an explicit integral pinching are spherical space forms. Precisely, we show that 3-dimensional Riemannian manifolds with totally geodesic boundary, positive scalar curvature and an explicit integral pinching between the $L^2$-norm of their scalar curvature and…

Paper proves rigidity of weak solutions for anisotropic N-Laplacian equations with Neumann or Robin boundary conditions.

The paper proves a new inequality for 3-manifolds with noncompact boundaries.

Study finite-type solutions of elliptic sinh-Gordon equation with Durham boundary conditions.

Defines distinguished curves for Poincaré-Einstein and singular geometries.

Unique solutions found for Plateau problems in smooth and continuous calibrations.

Compact spacelike hypersurface with constant curvature and boundary angles must be part of a hyperboloid.

We study realizations of pseudodifferential operators acting on sections of vector-bundles on a smooth, compact manifold with boundary, subject to conditions of Atiyah-Patodi-Singer type. Ellipticity and Fredholm property, compositions, adjoints and self-adjointness of such realizations are discussed. We construct regu…

Enhances neural network solvers for PDEs with complex boundary conditions.

Study proves symmetry of bounded domains in Riemannian manifolds.

Study proves radial symmetry in convex cones using subharmonic functions.

In this short note we outline a simple probabilistic proof of the Gauss-Bonnet formula for compact Riemannian manifolds with boundary, which adapts to this setting an argument due to Hsu \cite{Hs1,Hs2} in the closed case. The new technical ingredient is the Feynman-Kac formula for differential forms satisfying absolute…

The boundary-value problem for Laplace-type operators acting on smooth sections of a vector bundle over a compact Riemannian manifold with generalized local boundary conditions including both normal and tangential derivatives is studied. The condition of strong ellipticity of this boundary-value problem is formulated. …

In this paper we study asymptotically hyperbolic manifolds given as graphs of asymptotically constant functions over hyperbolic space $\bH^n$. The graphs are considered as subsets of $\bH^{n+1}$ and carry the induced metric. For such manifolds the scalar curvature appears in the divergence of a 1-form involving the int…

The Laplace-Beltrami operator (LBO) is a fundamental object associated to Riemannian manifolds, which encodes all intrinsic geometry of the manifolds and has many desirable properties. Recently, we proposed a novel numerical method, Point Integral method (PIM), to discretize the Laplace-Beltrami operator on point cloud…

Integral currents with boundary of finite mass are integral.

The paper establishes a correspondence between normal distributions and neat foliations on manifolds with boundary.

A new method calculates fractional moments using the moment-generating function.

Given a compact four dimensional smooth Riemannian manifold $(M,g)$ with smooth boundary, we consider the evolution equation by $Q$-curvature in the interior keeping the $T$-curvature and the mean curvature to be zero and the evolution equation by $T$-curvature at the boundary with the condition that the $Q$-curvature …

Study critical metrics on manifolds with boundary using integral and boundary estimates.

Study on heat flow across two half-lines with special boundary conditions.

We study the weighted integral transform on a compact manifold with boundary over a smooth family of curves $Γ$. We prove generic injectivity and a stability estimate under the condition that the conormal bundle of $Γ$ covers $T^*M$.

The simplicial volume of oriented closed connected smooth manifolds that admit a non-trivial smooth $S^1$-action vanishes. In the present work we prove a version of this result for the integral foliated simplicial volume of aspherical manifolds: The integral foliated simplicial volume of aspherical oriented closed conn…

Analyzes conformal anomaly in five dimensions, identifying new boundary conformal invariants.

Conditions for simple closed curves in surface covers.

In anomaly-free quantum field theories the integrand in the bosonic functional integral--the exponential of the effective action after integrating out fermions--is often defined only up to a phase without an additional choice. We term this choice ``setting the quantum integrand''. In the low-energy approximation to M-t…

We compute the integral homology of the space of paths in $\mathbb{C}P^n$ with endpoints in $\mathbb{R}P^n$, $n \ge 1$ and its algebra structure with respect to the Pontryagin-Chas-Sullivan product with $\mathbb{Z}/2$-coefficients.

Study rigidity of geodesic balls on manifolds with boundary.

BGNNs model particle-boundary interactions efficiently.

Paper solves a conjecture about minimal surfaces using sphere intersections and Weierstrass data.

Solves a problem posed by Brezis and Mironescu about least mass of area-minimizing currents.

The purpose of this paper is to study the validity of Stokes' Theorem for singular submanifolds and differential forms with singularities in Euclidean space. The results are presented in the context of Lebesgue Integration, but their proofs involve techniques from gauge integration in the spirit of R.~Henstock, J.~Kurz…

New BdryMatérn GP model for reliable boundary integration on irregular domains.

We prove that a critical metric of the volume functional on a $4$-dimensional compact manifold with boundary satisfying a second-order vanishing condition on the Weyl tensor must be isometric to a geodesic ball in a simply connected space form $\mathbb{R}^{4}$, $\mathbb{H}^{4}$ or $\mathbb{S}^{4}.$ Moreover, we provide…