arXiv research

We study Riemannian manifolds with boundary under a lower weighted Ricci curvature bound. We consider a curvature condition in which the weighted Ricci curvature is bounded from below by the density function. Under the curvature condition, and a suitable condition for the weighted mean curvature for the boundary, we ob…

Paper finds solutions to a complex equation on surfaces with boundary conditions.

Gradient estimate for harmonic functions with boundary condition proved.

Paper proves Faber-Krahn inequalities for weighted Laplacian eigenvalues.

Study solves Yamabe problems on metric measure spaces with or without boundary.

We consider a smooth Euclidean solid cone endowed with a smooth homogeneous density function used to weight Euclidean volume and hypersurface area. By assuming convexity of the cone and a curvature-dimension condition we prove that the unique compact, orientable, second order minima of the weighted area under variation…

New Dynkin condition for manifolds with boundary yields bi-Lipschitz equivalence and spectral properties.

No radial balanced metrics found on Kepler manifold unit ball with mild boundary conditions.

Paper proves a sharp weighted Isoperimetric inequality for substatic manifolds.

Combined with our previous work \cite{LW19eigenvalue}, we prove sharp lower bound estimates for the first nonzero eigenvalue of the weighted $p$-Laplacian with $1< p< \infty$ on a compact Bakry-Émery manifold $(M^n,g,f)$, without boundary or with a convex boundary and Neumann boundary condition, satisfying $\text{Ric}+…

Sharp heat equation gradient estimates on compact manifolds.

We prove regularity for a class of boundary value problems for first order elliptic systems, with boundary conditions determined by spectral decompositions, under coefficient differentiability conditions weaker than previously known. We establish Fredholm properties for Dirac-type equations with these boundary conditio…

The paper proves cohomology vanishing for a specific type of minimal submanifolds in a weighted Euclidean ball.

Paper proposes approximate Stein classes for efficient truncated density estimation.

We revisit the question of existence and regularity of minimizers to weighted least gradient problems on a fixed bounded domain, subject to a Dirichlet boundary condition, in the case where the boundary data is continuous and the weight function is C^2 and bounded away from zero. Under suitable geometric conditions on …

The paper proves smoothness of almost-minimizers' boundaries near the free boundary.

Given a compact Riemannian manifold (M, g) and two positive functions $ρ$ and $σ$, we are interested in the eigenvalues of the Dirichlet energy functional weighted by $σ$, with respect to the L 2 inner product weighted by $ρ$. Under some regularity conditions on $ρ$ and $σ$, these eigenvalues are those of the operator …

The paper finds upper bounds for eigenvalues of a weighted Laplacian.

The paper proves gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.

The aim of this note is to study the measure-valued Ricci tensor on smooth metric measure space with boundary, which is a generalization of Bakry-Emery's modified Ricci tensor on weighted Riemannian manifold. As an application, we offer a new approach to study curvature-dimension condition of smooth metric measure spac…

Sharp inequalities for radial functions on hyperbolic spaces without boundary conditions.

We prove weak and strong maximum principles, including a Hopf lemma, for smooth subsolutions to equations defined by linear, second-order, partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the smooth subsolutions along this boundary…

Lower bounds for Dirac eigenvalues on manifolds with boundary.

We discuss some consequences of the existence of the holomorphic quadratic Hopf differential on a conformally immersed constant mean curvature topological disc with analytic boundary. In particular, we derive a formula for the mean curvature as a weighted average of the normal curvature of the boundary curve, and a con…

It is known that by dualizing the Bochner-Lichnerowicz-Weitzenböck formula, one obtains Poincaré-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry-Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalize…

We study a Riemannian manifold equipped with a density which satisfies the Bakry--Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). We first obtain a Poincaré-type inequality on its boundary assuming that the latter is local…

Develops deep learning methods for non-linear PDEs in credit risk.

Our topological setting is a smooth compact manifold of dimension two or higher with smooth boundary. Although this underlying topological structure is smooth, the Riemannian metric tensor is only assumed to be bounded and measurable. This is known as a rough Riemannian manifold. For a large class of boundary condition…

Upper bounds on constants for Brownian motion with sticky boundary.

We study Riemannian manifolds with boundary under a lower $N$-weighted Ricci curvature bound for $N$ at most $1$, and under a lower weighted mean curvature bound for the boundary. We examine rigidity phenomena in such manifolds with boundary. We conclude a volume growth rigidity theorem for the metric neighborhoods of …

Analyzes Willmore flow for graphs with boundary data, proving existence and convergence.

Let $M$ be a weighted manifold with boundary $\partial M$, i.e., a Riemannian manifold where a density function is used to weight the Riemannian Hausdorff measures. In this paper we compute the first and the second variational formulas of the interior weighted area for deformations by hypersurfaces with boundary in $\p…

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 $T^*M$.

Study compares manifolds with boundary under weighted Ricci curvature bounds.

Generative adversarial networks (GANs) are a learning framework that rely on training a discriminator to estimate a measure of difference between a target and generated distributions. GANs, as normally formulated, rely on the generated samples being completely differentiable w.r.t. the generative parameters, and thus d…

The study examines stable regions in weighted manifolds with boundary properties.

The study characterizes hypersurfaces in weighted cylinders and generalizes confinement properties.

Under various elliptic boundary conditions, we obtain lower eigenvalue estimates for Dirac operators by using Hormander's weighted $L^2$-technique. Lower bounds in terms of the volume of the underlying manifolds are also deduced from the sharp Sobolev inequality due to Li and Zhu(\cite{LZ}).

Truncated densities are probability density functions defined on truncated domains. They share the same parametric form with their non-truncated counterparts up to a normalizing constant. Since the computation of their normalizing constants is usually infeasible, Maximum Likelihood Estimation cannot be easily applied t…

A new flow method solves the weighted Yamabe problem with boundary.

Given a compact four-dimensional Riemannian manifold $(M, g)$ with boundary, we study the problem of existence of Riemannian metrics on $M$ conformal to $g$ with prescribed $Q$-curvature in the interior $\mathring{M}$ of $M$, and zero $T$-curvature and mean curvature on the boundary $\partial M$ of $M$. This geometric …

In this paper we consider a class of weighted-volume preserving curvature flows acting on hypersurfaces that are trapped within two parallel hyperplanes and satisfy an orthogonal boundary condition. In the author's thesis the stability of cylinders under the flows was considered; it was found that they are stable provi…

We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with …

Sharp Steklov eigenvalue estimates for differential forms on manifolds.

Adapts Stein's method for geometric inequalities, addressing boundary terms.

Derives formulas for differential forms on weighted manifolds.

RFM improves CNFs by adding a boundary constraint term and matching velocity fields.

Study shows how to approximate and estimate high-dimensional classification functions without the curse of dimensionality.