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.
Study local structure of Einstein metrics with boundary conditions.
problem Understanding the local structure of Einstein metrics with boundary constraints.
method Analysis of moduli space of compact Einstein metrics, focusing on boundary conformal metric and mean curvature.
result For three dimensions, the map from Einstein metrics to boundary data is generically a local diffeomorphism.
We study the space of Riemannian metrics with positive scalar curvature on a compact manifold with boundary. These metrics extend a fixed boundary metric and take a product structure on a collar neighbourhood of the boundary. We show that the weak homotopy type of this space is preserved by certain surgeries on the bou…
Metric measure boundary vanishes on certain spaces without boundary.
problem Existence of infinite geodesics on Alexandrov spaces without boundary.
method Solving conjecture by showing metric measure boundary vanishes on mRCD(K,N) spaces. result Metric measure boundary vanishes on mRCD(K,N) spaces without boundary. The paper decomposes metrics on manifolds with boundaries.
problem Characterizing Riemannian metrics on manifolds with boundaries.
method Koiso-type decomposition and Ebin-type slice theorems.
result Characterization of relative Einstein metrics.
No radial balanced metrics found on Kepler manifold unit ball with mild boundary conditions.
problem Finding radial balanced metrics on the unit ball of the Kepler manifold.
method Analyzing boundary behavior and weights of metrics.
result Explicit weights for radial metrics satisfying balanced condition identified.
Boundary rigidity proven for non-reversible Finsler metrics.
problem Recovering non-reversible Finsler metrics from boundary distance data.
method Sum of reversible Finsler norm and closed 1-form, boundary rigidity results.
result 1-form can be uniquely recovered from boundary distance data.
Paper shows regions close to negatively curved metrics are minimal fillings and rigid.
problem Boundary rigidity and minimality of metrics near negatively curved ones.
method Generalizes previous work on filling volume minimality and boundary rigidity for almost hyperbolic metrics.
result Regions with metrics close to a negatively curved symmetric metric are strict minimal fillings and boundary rigid.
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 study examines metrics with unit volume or area on manifolds with boundaries, finding critical points and solving curvature problems.
problem Finding metrics with prescribed curvature on manifolds with boundaries.
method Variational properties of volume and boundary area functionals, using critical metrics and curvature conditions.
result Sufficient and necessary conditions for metrics to be critical points and for scalar/mean curvature functions.
The paper studies curvature conditions on manifolds with boundary.
problem Curvature preservation on manifolds with smooth boundaries.
method Constructing a family of metrics that agree with given metrics on the boundary and interior.
result Deforming metrics to ones with totally geodesic boundary while preserving curvature conditions.
Boundary distances determine conformal metrics
problem Determining conformal metrics from boundary distances
method Comparing renormalized boundary distances
result Metrics are equal if distances match
We investigate the validity of the isometry extension property for (Riemannian) Einstein metrics on manifolds with boundary. Given a metric on the boundary, this is the issue of whether any Killing field of the boundary metric extends to a Killing field of any bulk or filling Einstein metric inducing the given data on …
Constructs obstructions and deformation principles for positive scalar curvature metrics with mean convex boundaries.
problem Obstructing the existence of positive scalar curvature metrics with mean convex boundaries.
method Atiyah-Patodi-Singer index formula, deformation principle, homotopy equivalences, higher homotopy groups.
result Construction of compact manifolds with nontrivial higher homotopy groups for positive scalar curvature metrics with mean convex boundaries.
Defines timelike ideal boundary for non-positively curved Lorentzian spaces.
problem Understanding the geometry of non-positively curved Lorentzian spaces.
method Introduces timelike ideal boundary as asymptotic classes of geodesic rays, endows with topology and metric, and studies upper curvature bounds.
result Established upper curvature bounds for the resulting metric space.
Stokes equations help uniquely identify manifold metrics from boundary data.
problem Determining Riemannian metric from boundary Cauchy data.
method Proving uniqueness of metric from Stokes equations Cauchy data.
result Partial derivatives of all orders of the metric on the boundary are uniquely determined.
Study uniquely determines Riemannian metric derivatives from boundary data.
problem Determining Riemannian metric derivatives from boundary data.
method Computing the full symbol of the elastic Dirichlet-to-Neumann map.
result The elastic Dirichlet-to-Neumann map uniquely determines all partial derivatives of the Riemannian metric on the boundary.
Study solves Yamabe problems on metric measure spaces with or without boundary.
problem Yamabe-type problems on compact metric measure spaces with or without boundary.
method Analyzes uniqueness, characterization, and existence of minimizers.
result Characterizes weighted Yamabe solitons and existence of positive minimizers.
Compact metrics found with specific curvature properties on 3D surfaces.
problem Finding compact metrics with constant curvature on 3D surfaces.
method Blow-up analysis of Yamabe equation with critical Sobolev exponents.
result Proved the compactness of conformal metrics with constant scalar curvature and boundary mean curvature.
Rigidity theorem for special metrics on 4-manifolds.
problem Rigidity of Bach-flat metrics on manifolds with boundary.
method Critical point analysis of Weyl energy with boundary conditions.
result Rigidity of critical metrics on upper hemisphere.
Proves equivalence of two types of boundaries in metric spaces.
problem Proving equivalence of two types of boundaries in metric spaces.
method Analyzes and compares contracting and κ-Morse boundaries. result Proves equivalence of 1-Morse boundary and contracting boundary as topological spaces.
Paper examines stability of minimizing metrics on manifolds with boundary.
problem Stability of minimizing Yamabe metrics on compact manifolds with boundary.
method Investigates stability in the sense introduced by Escobar, showing closeness of nearly minimizing metrics.
result Quantitative closeness of nearly minimizing metrics to minimizing Yamabe metrics within their conformal class.
The paper explores conditions for positive scalar curvature on manifolds with boundaries and their doubles.
problem Conditions for positive scalar curvature on manifolds with boundaries and their doubles.
method Analyzes the relationship between boundary conditions and positive scalar curvature metrics on manifolds and their doubles.
result Provides conditions for positive scalar curvature metrics on manifolds with boundaries and their doubles.
New examples show positive scalar curvature metrics on manifolds with boundary that cannot be extended.
problem Positive scalar curvature metrics on manifolds with boundary that cannot be extended.
method Analytic techniques related to the prescribed scalar curvature problem in conformal geometry.
result Obstruction to positivity of conformal Laplacians given by a real-valued ξ-invariant.
Study of infinity in Teichmüller space using Thurston boundary.
problem Metric structure on Thurston boundary of Teichmüller space.
method Developed tools in sup metrics and applied Minsky's theorem.
result Investigation of metric structure on Thurston boundary.
Analytic torsion studied for fibred boundary metrics, with applications to conic degeneration.
problem Analytic torsion of fibred boundary metrics and conic degeneration.
method Established invariance and gluing formula for renormalized analytic torsion under deformations of metrics.
result Recovery of a result by Sher and Guillarmou about analytic torsion under conic degeneration.
We discuss the problem of prescribing the mean curvature and conformal class as boundary data for Einstein metrics on 3-manifolds, in the context of natural elliptic boundary value problems for Riemannian metrics.
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.
We obtain an Einstein metric of constant negative curvature given an arbitrary boundary metric in three dimensions, and a conformally flat one given an arbitrary conformally flat boundary metric in other dimensions. In order to compute the on-shell value of the gravitational action for these solutions, we propose to in…
We show for a non homogeneous boundary value problem for the Ricci flow on the disk that when the initial metric has positive curvature and the boundary is convex then the initial metric is deformed, via the normalized flow and along sequences of times, to a metric of constant curvature and totally geodesic boundary. W…
Proves existence of static vacuum metrics with specific boundary data.
problem Existence of static vacuum metrics with prescribed boundary data.
method Proves existence and local uniqueness of static vacuum metrics close to the Euclidean metric.
result Existence of static vacuum metrics with prescribed Bartnik boundary data.
Symplectic structures on Teichmüller spaces for surfaces with ideal boundary.
problem Understanding symplectic structures on Teichmüller spaces for surfaces with boundary.
method Defined natural symplectic structures and proved Hamiltonian properties.
result Explicit formula for Hill potential and global Darboux coordinates.
Proves well-posedness for Einstein equations with specific boundary conditions.
problem Well-posedness of vacuum Einstein equations with twisted Dirichlet boundary conditions.
method Proves local-in-time well-posedness for the IBVP of the Einstein equations with specified conformal class and scalar densities.
result Proves well-posedness for the Einstein equations with twisted Dirichlet boundary conditions.
We study the boundary rigidity problem for compact Riemannian manifolds with boundary (M,g): is the Riemannian metric g uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function ρg(x,y) known for all boundary points x and y? We prove in this paper global uniqueness …
In this paper, we study the regularity of asymptotically hyperbolic metrics with Einstein condition near boundary and Weyl curvature smooth enough in arbitrary dimension. Following Michael Anderson's method, we show that Cm,α conformally compact Riemannian metrics with Einstein equation vanishing to finite order n…
Study negative scalar curvature metrics with positive boundary mean curvature.
problem Bounding conformal metrics with specific curvature properties.
method Analyzing Riemannian manifolds with boundary conditions.
result A priori boundedness of metrics in specific cases.
Let (M,∂M) be a compact 3-manifold with boundary, which admits a convex co-compact hyperbolic metric. We consider the hyperbolic metrics on M such that the boundary is smooth and strictly convex. We show that the induced metrics on the boundary are exactly the metrics with curvature K>−1, and that the th…
In this paper we consider three-manifolds with weakly umbilic boundary (the Second Fundamental form of the boundary is a constant multiple of the metric). We show that if the initial manifold has positive Ricci curvature and the boundary is convex (nonnegative Second Fundamental form), its metric can be deformed via th…
A conformal metric on a 4-ball induces on the boundary 3-sphere a conformal metric and a trace-free second fundamental form. Conversely, such a data on the 3-sphere is the boundary of a unique selfdual conformal metric, defined in a neighborhood of the sphere. In this paper we characterize the conformal metrics and tra…
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.
The paper finds hyperbolic metrics on surfaces with boundary using combinatorial curvature flows.
problem Finding hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.
method Introducing combinatorial Ricci flow and combinatorial Calabi flow for generalized circle packings.
result Proves longtime existence and global convergence of combinatorial curvature flows.
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.
For spin manifolds with boundary we consider Riemannian metrics which are product near the boundary and are such that the corresponding Dirac operator is invertible when half-infinite cylinders are attached at the boundary. The main result of this paper is that these properties of a metric can be preserved when the met…
Rigidity theorem for curved manifolds with boundary.
problem Proving that certain curved manifolds are uniquely determined by their lens data.
method Analyzing lens data and using properties of Anosov type metrics.
result Local lens rigidity holds for negatively-curved manifolds and metrics of Anosov type.
A unique hyperbolic metric is found for each spherical cone-metric on the boundary of a hyperbolizable 3-manifold.
problem Finding a hyperbolic metric for a given spherical cone-metric on the boundary of a hyperbolizable 3-manifold.
method Constructing a strictly polyhedral hyperbolic metric on the 3-manifold such that the given spherical cone-metric is the induced dual metric on the boundary.
result The existence and uniqueness of a strictly polyhedral hyperbolic metric for a given spherical cone-metric on the boundary of a hyperbolizable 3-manifold.
The paper proves stability for scalar-flat metrics on manifolds with boundary.
problem Stability of scalar-flat metrics on manifolds with boundary.
method Reduced problem to boundary functional and used deficit control.
result Deficit controls distance to minimizing set on manifolds with boundary.
Solves a problem in Riemannian geometry for scalar-flat metrics with boundary conditions.
problem Finding a conformal metric with zero scalar curvature and prescribed boundary mean curvature.
method Construction of local test functions to resolve open cases and establish new solvability conditions.
result Established new solvability conditions for the problem.
The paper studies Ricci flow on manifolds with boundary, proving existence, uniqueness, and boundary conditions preservation.
problem Ricci flow on manifolds with boundary.
method Proving short-time existence and uniqueness of the solution, and showing boundary conditions preservation.
result The flow preserves natural boundary conditions under certain curvature conditions.