arXiv research

Introduce new boundary mass for asymptotically flat half-manifolds

We consider two cases of the asymptotically flat scalar-flat Yamabe problem on a non-compact manifold with boundary, in dimension $n\geq3$. First, following arguments of Cantor and Brill in the compact case, we show that given an asymptotically flat metric $g$, there is a conformally equivalent asymptotically flat scal…

Rigidity theorem for ideal surfaces with flat boundary conditions.

Study the mass of flat 3-manifolds with boundary using specific methods.

Path connectedness of boundaries for certain CAT(0) groups with isolated flats.

Local conditions on boundaries of $C^\infty$ Levi-flat hypersurfaces, in case the boundary is a generic submanifold, are studied. For nontrivial real analytic boundaries we get an extension and uniqueness result, which forces the hypersurface to be real analytic. This allows us to classify all real analytic generic bou…

The paper proves compactness of scalar-flat metrics on manifolds with umbilic boundary.

Classifies stability of flat-core $p$-elasticae pinned at boundaries.

Proves Penrose inequality for specific asymptotically flat manifolds.

The paper proves compactness of scalar-flat metrics on low-dimensional manifolds with umbilic boundary.

The paper proves stability for scalar-flat metrics on manifolds with boundary.

The study extends a theorem to bundles on manifolds with boundaries.

Positive mass theorem for non-smooth metrics on flat manifolds with corners.

We solve the problem on flat extensions of a generic surface with boundary in Euclidean 3-space, relating it to the singularity theory of the envelope generated by the boundary. We give related results on Legendre surfaces with boundaries via projective duality and observe the duality on boundary singularities. Moreove…

Every link is shown to be presentable as a boundary of an unknotted flat banded surface. A (flat) banded link is defined as a boundary of an unknotted (flat) banded surface. A link's (flat) band index is defined as the minimum number of bands required to present the link as boundaries of an unknotted (flat) banded surf…

Constructs metrics with negative curvature on specific manifold types.

Characterizes hypergenerated stratified groups with flat boundaries.

We give global restrictions on the possible boundaries of compact, orientable, locally conformally flat manifolds of dimension $4k$ in terms of integrality of eta invariants.

We establish a gluing theorem for solutions of a Yamabe problem for manifolds with boundary studied by Escobar in the 90's. Given two scalar-flat Riemannian manifolds whose boundary has zero mean curvature and sharing a submanifold $K$, we produce the generalized connected sum along $K$. On this third manifold we produ…

Solves a problem in Riemannian geometry for scalar-flat metrics with boundary conditions.

Let (M,g) be a compact Riemannian manifold with boundary. This paper addresses the Yamabe-type problem of finding a conformal scalar-flat metric on M, which has the boundary as a constant mean curvature hypersurface. When the boundary is umbilic, we prove an existence theorem that finishes some remaining cases of this …

Study shows convergence of volumes on manifolds with boundary under area constraints.

We prove a positive mass theorem for $n$-dimensional asymptotically flat manifolds with a non-compact boundary if either $3\leq n\leq 7$ or if $n\geq 3$ and the manifold is spin. This settles, for this class of manifolds, a question posed in a recent paper by the first author in connection with the long-term behavior o…

Rigidity theorem for special metrics on 4-manifolds.

Let (M,g) be a compact Riemannian three-dimensional manifold with boundary. We prove the compactness of the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. This involves a blow-up analysis of a Yamabe-type equation with critical Sobolev e…

Paper proves rigidity and index of Y-cones in unit ball.

Flat minimal hypersurfaces found in wedge-shaped domains.

The purpose of this article is to investigate Bach-flat critical metrics of the volume functional on a compact manifold $M$ with boundary $\partial M.$ Here, we prove that a Bach-flat critical metric of the volume functional on a simply connected 4-dimensional manifold with boundary isometric to a standard sphere must …

In this paper we establish existence and compactness of solutions to a general fully nonlinear version of the Yamabe problem on locally conformally flat Riemannian manifolds with umbilic boundary.

We show that the recently introduced L1TV functional can be used to explicitly compute the flat norm for co-dimension one boundaries. While this observation alone is very useful, other important implications for image analysis and shape statistics include a method for denoising sets which are not boundaries or which ha…

In this paper we prove that every Riemannian metric on a locally conformally flat manifold with umbilic boundary can be conformally deformed to a scalar flat metric having constant mean curvature. This result can be seen as a generalization to higher dimensions of the well known Riemann mapping Theorem in the plane.

The classical knot groups are the fundamental groups of the complements of smooth or piecewise-linear (PL) locally-flat knots. For PL knots that are not locally-flat, there is a pair of interesting groups to study: the fundamental group of the knot complement and that of the complement of the ``boundary knot'' that occ…

We study a conformal flow for compact Riemannian manifolds of dimension greater than two with boundary. Convergence to a scalar-flat metric with constant mean curvature on the boundary is established in dimensions up to seven, and in any dimensions if the manifold is spin or if it satisfies a generic condition.

Let (M,g) be a compact Riemannian manifold with boundary. This paper is concerned with the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We prove that this set is compact for dimensions greater than or equal to 7 under the generic condi…

Consider a closed marked flat surface $S$ of genus $g\geq 2$ and area 1 and its universal covering $\tilde{S}$. We show that the measure class of the Hausdorff measure of the Gromov boundary of $\tilde{S}$ uniquely determines $S$.

Inspired by the work of F. Hang and X. Wang and partial results by S. Raulot, we prove a scalar curvature rigitidy result for locally conformally flat manifolds with boundary in the spirit of the well-known Min-Oo conjecture.

We consider the volume entropy of closed flat surfaces of genus $g\geq 2$ and area 1. We show that a sequence of flat surfaces diverges in the moduli space if and only if the volume entropy converges to infinity. Equivalently the Hausdorff dimension of the Gromov boundary of the isometric universal cover tends to infin…

Unified rigidity theorem for Plateau surfaces in $\mathbb{B}^n$.

The study proves a new upper bound for isoperimetric ratio in scalar-flat conformal classes.

The paper proves stability of manifolds with boundary under volume and distance constraints.

Paper proves flatness of anisotropic minimal graphs in half-spaces.

The study proves rigidity for Poincaré-Einstein manifolds with flat Euclidean conformal infinity.

Let (M,g) be a compact n-dimensional Riemannian manifold with boundary. This article is concerned with the set of scalar-flat metrics on M which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We construct examples of metrics on the unit ball, in dimensions n>=25, for wh…

We address the problem of existence and uniqueness of a Levi-flat hypersurface $M$ in $C^n$ with prescribed compact boundary $S$ for $n\ge3$. The situation for $n\ge3$ differs sharply from the well studied case $n=2$. We first establish necessary conditions on $S$ at both complex and CR points, needed for the existence…

We extend the idea and techniques in \cite{Miao} to study variational effect of the boundary geometry on the ADM mass of an asymptotically flat manifold. We show that, for a Lipschitz asymptotically flat metric extension of a bounded Riemannian domain with quasi-convex boundary, if the boundary mean curvature of the ex…

We consider the scattering and lens rigidity of compact surfaces with boundary that have a trapped geodesic. In particular we show that the flat cylinder and the flat Möbius strip are determined by their lens data. We also see by example that the flat Möbius strip is not determined by it's scattering data. We then cons…

Improved mass-capacity bounds for specific 3D manifolds.

Paper unifies three invariants for flat bundles over surfaces with boundary.