Unique compact Fuchsian manifolds with convex boundary are determined by their boundary.
problem Identifying compact Fuchsian manifolds with convex boundaries.
method Proving uniqueness based on the induced path metric on the boundary.
result Compact Fuchsian manifolds with convex boundaries are uniquely determined by the induced path metric on the boundary.
Study on quasi-Einstein manifolds with boundary estimates and inequalities.
problem Understanding the geometry of compact quasi-Einstein manifolds with boundary.
method Sharp boundary estimates and characterization theorems for quasi-Einstein manifolds.
result New geometric inequalities and boundary estimates for quasi-Einstein manifolds.
Study characterizes compact Einstein-type manifolds with boundary.
problem Characterize compact Einstein-type manifolds with nonempty boundary.
method Proved a sharp boundary estimate, obtained Hawking mass bounds, and provided a topological classification for the boundary.
result Obtained a gap result for compact Einstein-type manifolds with boundary.
The goal of this article is to study compact quasi-Einstein manifolds with boundary. We provide boundary estimates for compact quasi-Einstein manifolds simi\-lar to previous results obtained for static and V-static spaces. In addition, we show that compact quasi-Einstein manifolds with connected boundary and satisfyi…
Study critical metrics on manifolds with boundary using integral and boundary estimates.
problem Investigate geometry of critical metrics on compact manifolds with boundary.
method Use generalized Reilly's formula to derive integral and boundary estimates.
result Establish new boundary estimates for critical metrics of the volume functional.
Proves multiplicity one for boundary minimal hypersurfaces in compact manifolds.
problem Proving multiplicity one for min-max free boundary minimal hypersurfaces in compact manifolds with boundary.
method Developed existence and regularity theory for free boundary hypersurfaces with prescribed mean curvature, including Morse index bounds.
result Proved multiplicity one theorem for min-max free boundary minimal hypersurfaces in compact manifolds with boundary.
The paper proves compactness of scalar-flat metrics on low-dimensional manifolds with umbilic boundary.
problem Finding scalar-flat metrics with specific boundary conditions.
method Analyzing compact Riemannian manifolds with umbilic boundaries and proving compactness of scalar-flat metrics under certain conditions.
result Scalar-flat metrics are a compact set in low-dimensional manifolds (n=6,7,8) when the Weyl tensor is non-zero on the boundary.
Study rigidifies Einstein-type manifolds with boundary and constant curvature.
problem Classifying compact Einstein-type manifolds with boundary and constant scalar curvature.
method Applied recent results on gradient Einstein-type manifolds to prove rigidity.
result Rigidity results for compact Einstein-type manifolds with boundary and constant scalar curvature.
Generalized stability theorem for compact manifolds with boundary.
problem Stability of manifolds with boundary.
method Equivariant μ-bubbles technique.
result Yamabe invariant of compact manifolds with boundary is positive if and only if the invariant of the manifold times S^1 is positive.
Positive mass theorem for non-smooth metrics on flat manifolds with corners.
problem Proving a positive mass theorem for non-smooth metrics on asymptotically flat manifolds with non-compact boundary.
method Proves a positive mass theorem for metrics that are only continuous across a compact hypersurface.
result Obtains a positive mass theorem on manifolds with non-compact corners.
Extends Llarull's theorem to noncompact manifolds with boundary.
problem Generalizing Llarull's theorem to noncompact manifolds with boundary.
method Extends previous results to include compact boundaries.
result Generalized theorem to noncompact manifolds with compact boundaries.
We compute the index of a Callias-type operator with APS boundary condition on a manifold with compact boundary in terms of combination of indexes of induced operators on a compact hypersurface. Our result generalizes the classical Callias-type index theorem to manifolds with compact boundary.
The paper proves conditions for smooth convergence of hyperkaehler 4-manifolds with boundary.
problem Compactness of hyperkaehler 4-manifolds with boundary.
method Analyzes sequences of hyperkaehler triples under topological and curvature conditions.
result Smooth convergence of hyperkaehler triples up to diffeomorphisms if boundary restrictions converge.
Compact manifolds with boundary have finite type mapping class groups.
problem Understanding the structure of mapping class groups for manifolds with boundaries.
method Proved finite type for mapping class groups under specific dimension and connectivity assumptions.
result Mapping class groups of compact manifolds with boundaries are of finite type.
Unique hyperbolic manifolds identified by boundary pleating.
problem Identifying hyperbolic manifolds from their boundary pleating.
method Used pleating measured lamination on the boundary of convex cores.
result Convex co-compact hyperbolic manifolds are uniquely determined by their pleating lamination.
Study on compactness and blow-up of solutions for Yamabe problems on manifolds with non-umbilic boundaries.
problem Compactness and blow-up behavior of solutions to the Yamabe boundary problem on manifolds with non-umbilic boundaries.
method Analysis of stability and blow-up sequences for solutions under perturbations of mean curvature and scalar curvature.
result Existence of a blowing-up sequence of solutions when perturbing the mean curvature from above or below with a function having a large positive maximum.
In 1960s, Almgren initiated a program to find minimal hypersurfaces in compact manifolds using min-max method. This program was largely advanced by Pitts and Schoen-Simon in 1980s when the manifold has no boundary. In this paper, we finish this program for general compact manifold with nonempty boundary. As a result, w…
Paper proves Luo's conjecture for 3D triangulated manifolds.
problem Finding hyperbolic metrics on compact 3-manifolds with boundary.
method Introduced and extended combinatorial Ricci flow to handle singularities.
result Proved Luo's conjecture affirmatively for ideal triangulations.
New characterizations for manifolds with boundary rigidity results.
problem Rigidity results for compact gradient Einstein-type manifolds with boundaries.
method Analyzing manifolds with rigidity results and characterizations.
result New topological and geometric characterizations for manifolds with boundaries.
Study of elliptic boundary value problems on non-compact manifolds.
problem Analyzing elliptic differential operators on manifolds with non-compact boundaries.
method Regularity theory and trace theorems for sections in the maximal domain under various assumptions.
result Systematic study of local and nonlocal boundary conditions, including the Atiyah-Patodi-Singer condition.
We study the space of smooth Riemannian structures on compact three-manifolds with boundary that satisfies a critical point equation associated with a boundary value problem, for simplicity, Miao-Tam critical metrics. We provide an estimate to the area of the boundary of Miao-Tam critical metrics on compact three-manif…
Develops a method to deform metrics on manifolds with non-compact boundaries.
problem Creating metrics with positive scalar curvature on manifolds with boundary.
method General deformation principle for Riemannian metrics on manifolds with non-compact boundaries.
result Non-existence of metrics with positive scalar curvature and mean convex boundary.
Constructs Poisson structures with compact support on manifolds.
problem Creating Poisson structures with compact support on manifolds.
method Explicit construction of Poisson structures with polynomial coefficients and modification outside open balls.
result Even-dimensional manifolds can be equipped with Poisson structures that vanish to infinite order at codimension one subsets.
Study of prescribing scalar curvature and mean curvature on compact manifolds with boundary.
problem Prescribing scalar curvature and mean curvature on compact manifolds with boundary.
method Introducing singular metrics inspired by previous work on closed manifolds, proving rigidity results for flat manifolds with totally geodesic boundary.
result Generic scalar-flat manifolds with minimal boundary can have scalar curvature and mean curvature prescribed simultaneously.
Paper reconstructs compact Riemannian manifolds from travel time data.
problem Reconstructing compact Riemannian manifolds from partial travel time data.
method Embedding in function space, studying distance function regularity.
result Reconstruction of compact Riemannian manifolds from travel time data.
The study classifies quasi-Einstein manifolds with boundary.
problem Classifying quasi-Einstein manifolds with boundary.
method Analyzing scalar curvature and isometric properties.
result Compact quasi-Einstein manifolds with boundary are rigid.
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…
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…
The study of rigidity theorems on 4-manifolds with boundary.
problem Understanding topological restrictions on 4-manifolds with boundary.
method Introducing new conformal and smooth invariants, studying Weyl functional, and analyzing the expansion of a smooth Riemannian metric near the boundary.
result Established several conformally invariant rigidity theorems for 4-manifolds with boundary.
The paper proves the existence of boundary minimal hypersurfaces in compact manifolds with boundary.
problem Existence of boundary minimal hypersurfaces in compact manifolds with boundary.
method Min-max theory applied to local maximizers of width in conformal classes.
result Existence of a sequence of properly embedded equidistributed boundary minimal hypersurfaces.
Solves Einstein constraint equations on compact manifolds with specified boundaries.
problem Solving Einstein constraint equations with specified boundaries.
method Studies conformal constraint equations with low regularity assumptions.
result Solves Einstein constraint equations on compact manifolds with specified boundaries.
Classifies 4-manifolds with elementary amenable groups and their boundaries.
problem Characterizing compact aspherical 4-manifolds with elementary amenable fundamental groups.
method Classification based on fundamental group properties and Farrell-Jones Conjecture.
result Such manifolds are either polycyclic or solvable Baumslag-Solitar groups.
We consider the product of a compact Riemannian manifold without boundary and null scalar curvature with a compact Riemannian manifold with boundary, null scalar curvature and constant mean curvature on the boundary. We use bifurcation theory to prove the existence of a infinite number of conformal classes with at leas…
Proves bounded subsolution theorem for complex Monge-Ampère equation on compact Hermitian manifolds.
problem Complex Monge-Ampère equation with positive Radon measure on compact Hermitian manifolds.
method Proves bounded subsolution theorem.
result Establishes bounded subsolution theorem for complex Monge-Ampère equation.
We prove a positive mass theorem for n-dimensional asymptotically flat manifolds with a non-compact boundary if either 3≤n≤7 or if n≥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…
Curvature conditions distinguish Euclidean space and disks in contractible manifolds.
problem Distinguish Euclidean space and disks among contractible manifolds.
method Investigate curvature conditions on open and compact contractible manifolds with boundary.
result Stronger curvature conditions can distinguish disks from Euclidean spaces.
We prove that knowing the length of geodesics joining points on the boundary of a two-dimensional, compact, simple Riemannian manifold with boundary, we can determine uniquely the Riemannian metric up to the natural obstruction.
Study on electrostatic systems with boundary, proving new geometric inequalities.
problem Electrostatic systems with boundary in higher dimensions.
method Investigation of electrostatic systems on compact manifolds with boundary, establishing new geometric properties.
result Proved sharp boundary estimates and isoperimetric-type inequalities for electrostatic manifolds.
In this note, we extend the Bridgeman-Kahn identity to all finite-volume orientable hyperbolic n-manifolds with totally geodesic boundary. In the compact case, Bridgeman and Kahn are able to express the manifold's volume as the sum of a function over only the orthospectrum. For manifolds with non-compact boundary, ou…
A triangulation of a compact 3-manifold is annular-efficient if it is 0-efficient and the only normal, incompressible annuli are thin edge-linking. If a compact 3-manifold has an annular-efficient triangulation, then it is irreducible, boundary-irreducible, and an-annular. Conversely, it is shown that for a compact, ir…
We analyze critical points of two functionals of Riemannian metrics on compact manifolds with boundary. These functionals are motivated by formulae of the mass functionals of asymptotically flat and asymptotically hyperbolic manifolds.
Sharp heat equation gradient estimates on compact manifolds.
problem Gradient estimates for positive solutions on weighted manifolds.
method Proving sharp gradient estimates for positive solutions to the weighted heat equation.
result Refined gradient estimates and Liouville theorems for ancient solutions.
3D spaces without boundaries are found that aren't manifolds.
problem Finding compact aspherical Alexandrov spaces without boundaries.
method Constructing specific examples of 3D spaces.
result Examples of 3D spaces without boundaries that are not topological manifolds.
Study finds conditions for metrics on curved spaces.
problem Finding metrics with specific curvature properties.
method Analyzes a conformal class with prescribed scalar and boundary mean curvatures.
result Establishes a necessary and sufficient condition involving a conformal invariant.
We show how to construct absolutely exotic smooth structures on compact 4-manifolds with boundary, including contractible manifolds. In particular, we prove that any compact smooth 4-manifold W with boundary that admits a relatively exotic structure contains a pair of codimension-zero submanifolds homotopy equivalent t…
The paper proves infinitely many free boundary minimal hypersurfaces in compact manifolds.
problem Existence of free boundary minimal hypersurfaces in compact Riemannian manifolds.
method Adaptations of A. Song's work and Marques-Neves' resolution to Yau's conjecture, combined with Li-Zhou's regularity theorem.
result Proves the existence of infinitely many almost properly embedded free boundary minimal hypersurfaces in compact manifolds.
Constructs conformal metrics with negative curvature on manifolds with boundary.
problem Creating conformal metrics with negative curvature on manifolds with boundary.
method Using Morse functions to construct conformal metrics and proving results for compact 3-manifolds with boundary.
result Any Riemannian metric on compact 3-manifolds with boundary is conformal to a compact metric of negative sectional curvature.
The article classifies curvature functions on compact manifolds with boundaries.
problem Prescribing scalar and mean curvature functions on compact manifolds with boundaries.
method Classification based on the sign of the first eigenvalue of the conformal Laplacian.
result A 'Trichotomy Theorem' for curvature functions is established.