arXiv research

Estimating boundaries from point clouds with improved accuracy and rigorous error estimates.

This paper reverses a construction by merging boundary critical points into an interior one.

Minimal surfaces' boundary points are always smooth.

Simply connected surfaces with large constant mean curvature and free boundaries concentrate at critical points of the boundary's mean curvature.

Develops a method for conformal parameterization of point clouds without fixed boundaries.

In [8] the authors introduced a pair of new de Rham complexes on a compact oriented Riemannian manifold with boundary by using a pair of new boundary conditions to discuss the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the Lefschetz fixed point formula on these complexes with…

We construct non-trapping asymptotically hyperbolic manifolds with boundary conjugate points but no interior conjugate points.

Maps with boundary definite fold points restrict manifold structure.

Critical hypersurfaces with boundary have unique shapes and properties.

Rigidity for 4D Willmore submanifolds with boundary.

New mathematical surfaces without boundaries found.

The Jacobi-Maupertuis metric allows one to reformulate Newton's equations as geodesic equations for a Riemannian metric which degenerates at the Hill boundary. We prove that a JM geodesic which comes sufficiently close to a regular point of the boundary contains pairs of conjugate points close to the boundary. We prove…

We study the boundary rigidity problem with partial data consisting of determining locally the Riemannian metric of a Riemannian manifold with boundary from the distance function measured at pairs of points near a fixed point on the boundary. We show that one can recover uniquely and in a stable way a conformal factor …

We determine the set of Busemann points of an arbitrary finite-dimensional normed space. These are the points of the horofunction boundary that are the limits of "almost-geodesics". We prove that all points in the horofunction boundary are Busemann points if and only if the set of extreme sets of the dual unit ball is …

We prove that tangent cones at singular boundary points of a two-dimensional current almost area minimizing are unique. Following the ideas exposed by White in [8], the result is achieved by combining a suitable epiperimetric inequality and an almost-monotonicity formula for the mass at boundary points.

A new boundary for geodesic spaces defined and studied.

New boundary and point constraints for controlling conformal surfaces.

We prove a sharp version of the Hopf boundary point lemma for Black-Scholes type equations. We also investigate the existence and the regularity of the spatial derivative of the solutions at the spatial boundary.

New boundary constructed for mapping class group.

The paper improves boundary detection and density estimation on noisy data.

The paper studies Morse flows on 3-manifold boundaries with fixed points.

Study shows unique tangent cones for area-minimizing currents at boundary points.

The paper explores geometric properties of free boundary hypersurfaces in balls.

We develop Morse theory for manifolds with boundary. Besides standard and expected facts like the handle cancellation theorem and the Morse lemma for manifolds with boundary, we prove that, under a topological assumption, a critical point in the interior of a Morse function can be moved to the boundary, where it splits…

The study shows that certain graphs are regular at boundary points.

We prove a local boundary regularity result for the complete Kahler-Einstein metrics of negative Ricci curvature near strictly pseudoconvex boundary point. We also study the asymptotic behaviour of their holomorphic bisectional curvatures near such points.

Study on Santaló point for convex bodies in normed spaces.

The paper establishes a version of the Hopf boundary point lemma for sections of a vector bundle over a manifold with boundary. This result may be viewed as a counterpart to the tensor maximum principle obtained by R. Hamilton in 1986. Potential applications include the study of various geometric flows and the construc…

The paper derives Gauss-Bonnet formulas for mappings between surfaces with boundary.

Study on the geometry of limit spaces of manifolds with boundary.

The n-strand braid group can be defined as the fundamental group of the configuration space of n unlabeled points in a closed disk based at a configuration where all n points lie in the boundary of the disk. Using this definition, the subset of braids that have a representative where a specified subset of these points …

Optimal geodesics connect boundary points in Teichmüller space.

Study Zoll manifolds with boundary, showing unique geodesic properties.

The study examines metrics with unit volume or area on manifolds with boundaries, finding critical points and solving curvature problems.

Proves involutions on Right-angled Coxeter groups without fixed points.

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…

Let X be a compact 2-manifold with nonempty boundary dX and let f: (X, dX) --> (X, dX) be a boundary-preserving map. Denote by MF_d[f] the minimum number of fixed point among all boundary-preserving maps that are homotopic through boundary-preserving maps to f. The relative Nielsen number N_d(f) is the sum of the numbe…

We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of $C^{1,α}$ submanifolds, possibly with multiplicity, meeting tangentially, given that the current has a tangent cone supported in a hyperplane with constant orientation vector; t…

This paper focuses on the problem of topological equivalence of functions with isolated critical points on the boundary of a compact surface $M$ which are also isolated critical points of their restrictions to the boundary. This class of functions we denote by $Ω(M)$. Firstly, we've obtained the topological classificat…

We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of $C^{1,α}$ submanifolds, possibly with multiplicity, meeting tangentially, given that the current has a tangent cone supported in a hyperplane with constant orientation vector; t…

We construct new examples of complete Einstein metrics on balls. At each point of the boundary at infinity, the metric is asymptotic to a homogeneous Einstein metric on a solvable group, which varies with the point at infinity.

Study shows Bergman metric is non-Einstein for certain domains.

Study of group boundaries and subgroup properties.

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

The paper classifies vertices in planar polygons formed by convex domains.

In this paper, we investigate the structure of the Gardiner-Masur boundary of Teichmuller space. Indeed, we will give a geometric description of boundary comparing to the Duchin-Leininger-Rafi compactification of the space of singular flat structures. We will obtain the coincidence between the Gardiner-Masur boundary a…

In this paper, we shall show that the metric boundary of the Teichmueller space with respect to the Teichmueller distance contains non-Busemann points when the complex dimension of the Teichmueller space is at least two.

Deep learning models have been the subject of study from various perspectives, for example, their training process, interpretation, generalization error, robustness to adversarial attacks, etc. A trained model is defined by its decision boundaries, and therefore, many of the studies about deep learning models speculate…