The study quantifies geodesic divergence on Riemannian planes with bounded geometry.
problem Understanding geodesic divergence on Riemannian planes with specific geometric constraints.
method Recalling quasi-redirection and using it to quantify geodesic divergence, compactifying Riemannian planes into D2 or S2. result Necessary and sufficient conditions for the quasi-redirecting compactification being S2 are derived in terms of asymptotic cones. The paper studies topological and dynamic properties of boundaries in geometric group actions.
problem Understanding the topological and dynamic properties of boundaries in geometric group actions.
method Developed and studied sublinearly Morse and quasi-redirecting boundaries for proper geodesic spaces with geometric group actions.
result Proved that the action of a group on the boundaries is minimal and that the boundaries are topological spaces.
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.
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 introduce new boundary conditions for differential forms on symplectic manifolds with boundary. These boundary conditions, dependent on the symplectic structure, allows us to write down elliptic boundary value problems for both second-order and fourth-order symplectic Laplacians and establish Hodge theories for the …
Abstracts a construction of boundary triplets for self-adjoint elliptic problems.
problem Computing the index of families of self-adjoint elliptic boundary problems.
method Abstract axiomatic version of boundary triplets and their applications.
result Analytic proof of index theorem and computation of index differences.
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.
We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. The adapted boundary operator need not be selfadjoint and the boundary condition need not be pseudo-local. We show the equivalence of various characterisations of elliptic boundary conditions and demonst…
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.
Generalizes Bestvina's Z-boundaries to coarse Z-boundaries.
problem Establishing properties of Z-boundaries for groups. method Introducing a new concept of a 'coarse Z-boundary' and proving theorems about it. result Admitting a coarse Z-boundary is a pure quasi-isometry invariant. 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.
We introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with rays that identify the "hyperbolic directions" in that space. This boundary is a quasi-isometry invariant and thus produces a well-defined boundary for any finitely generated group. In the case of a prope…
Foundations for free boundary Brakke flows established.
problem Analyzing free boundary flows through singularities
method Introducing unit-regular and cyclic free boundary flows, proving avoidance principle, and introducing inner and outer flows.
result General tools for analyzing free boundary flows through singularities.
Proof of local well-posedness for a specific boundary condition in general relativity.
problem Initial boundary value problem in general relativity with umbilic boundary condition.
method Wave coordinates and key observation of momentum constraint validity for umbilic boundaries.
result Local well-posedness established for the initial boundary value problem.
Study finds rigid properties of boundary-free hypersurfaces in specific data sets.
problem Rigidity of free boundary hypersurfaces in initial data sets with boundary.
method Extending local splitting theorems and applying results on free boundary MOTS.
result Rigidity results for compact free boundary hypersurfaces in initial data sets with boundary.
The paper studies g-stability of surfaces with boundary and derives area estimates.
problem Investigating g-stability of surfaces with boundary. method Analyzing geometric properties and deriving area estimates.
result Derives area estimates and determines the topology of the surface.
The paper shows how sublinearly Morse boundaries can be understood through combinatorial methods.
problem Understanding sublinearly Morse boundaries in cubulated groups and CAT(0) cube complexes.
method Combining geometric and combinatorial approaches to analyze sublinearly Morse boundaries.
result Sublinearly Morse boundaries can be described combinatorially and continuously related to Gromov and Roller boundaries.
Homotopy equivalent boundaries of cube complexes are studied.
problem The equivalence of different boundaries of cube complexes.
method Using a partial order on a quotient of the Roller boundary, we obtain the simplicial Roller boundary and show homotopy equivalence among the Tits, simplicial, and simplicial Roller boundaries.
result The Tits, simplicial, and simplicial Roller boundaries are homotopy equivalent.
New examples of non-smoothable homeomorphisms of 4-manifolds with boundary found.
problem Finding non-smoothable homeomorphisms of 4-manifolds with boundary.
method Constructing specific examples of homeomorphisms that fix the boundary and act trivially on homology.
result First examples of non-smoothable self-homeomorphisms of smooth 4-manifolds with boundary.
To every Gromov hyperbolic space X one can associate a space at infinity called the Gromov boundary of X. Gromov showed that quasi-isometries of hyperbolic metric spaces induce homeomorphisms on their boundaries, thus giving rise to a well-defined notion of the boundary of a hyperbolic group. Croke and Kleiner showed t…
We present an introduction to boundary value problems for Dirac-type operators on complete Riemannian manifolds with compact boundary. We introduce a very general class of boundary conditions which contains local elliptic boundary conditions in the sense of Lopatinskij and Shapiro as well as the Atiyah-Patodi-Singer bo…
The paper classifies algebraic curves in 4-balls and their boundaries.
problem Understanding algebraic curves in 4-dimensional balls and their boundaries.
method Analyzing algebraic curves in complex 2-space and their intersections with 4-balls.
result Classification of algebraic curves with up to 5 crossings.
Manifolds uniquely identified by boundary distance differences.
problem Identifying Riemannian manifolds by their boundary distances.
method Distance difference representation on non-convex boundaries without restrictions.
result Complete Riemannian manifolds uniquely determined by their boundary distances.
We establish a moduli space E of stationary vacuum metrics in a spacetime, and set up a well-defined boundary map Π in E, assigning a metric class with its Bartnik boundary data. Furthermore, we prove the boundary map Π is Fredholm by showing that the stationary vacuum equations (combined with p…
We define a class of boundary value problems on manifolds with fibered boundary. This class is in a certain sense a deformation between the classical boundary value problems and the Atiyah-Patodi-Singer problems in subspaces. The boundary conditions in this theory are taken as elements of the C^*-algebra generated by p…
We study boundary value problems for the Dirac operator on Riemannian Spinc manifolds of bounded geometry and with noncompact boundary. This generalizes a part of the theory of boundary value problems by C. Bär and W. Ballmann for complete manifolds with closed boundary. As an application, we derive the lower bound …
Classifies local boundary conditions for Dirac-type operators on manifolds.
problem Determining all local smooth boundary conditions for Dirac-type operators.
method Combining general theory of boundary value problems for Dirac operators and pointwise considerations.
result Classification of local self-adjoint regular boundary conditions for Dirac spinors in dimensions 3 and 4.
In this paper, we consider a concentration of measure problem on Riemannian manifolds with boundary. We study concentration phenomena of non-negative 1-Lipschitz functions with Dirichlet boundary condition around zero, which is called boundary concentration phenomena. We first examine relation between boundary concen…
Minimal surfaces' boundary points are always smooth.
problem Boundary regularity of minimal surfaces.
method Proving all boundary points are regular submanifolds.
result Boundary points of minimal surfaces are regular.
Symplectic structure found on projective structures on surfaces with boundary.
problem Deformation space of projective structures on surfaces with boundary.
method Natural symplectic structure on the space, integrating the Adler-Gelfand-Dikii-space of the boundary.
result Space is a Hamiltonian space for the symplectic groupoid.
Constructs minimal surfaces near the boundary of a ball.
problem Creating minimal surfaces close to the boundary of a ball.
method PDE gluing methods to construct FBMS of genus zero.
result Desingularizations of catenoidal annuli and flat discs near the boundary.
In this contribution we derive an explicit formula for the boundary non-crossing probabilities for Slepian processes associated with the piecewise linear boundary function. This formula is used to develop an approximation formula to the boundary non-crossing probabilities for general continuous boundaries. The formulas…
Study on the geometry of limit spaces of manifolds with boundary.
problem Understanding the geometry of limit spaces of manifolds with boundary.
method Developed infinitesimal geometry for limit spaces under curvature and diameter bounds.
result Determined the infinitesimal structure and Hausdorff dimensions of boundary singular sets.
For a compact Riemannian manifold with boundary, we want to find the metric structure from knowledge of distances between boundary points. This is called the "boundary rigidity problem". If the boundary is not concave, which means locally not all shortest paths lie entirely in the boundary, then we are able to find the…
Study Zoll manifolds with boundary, showing unique geodesic properties.
problem Characterize Zoll manifolds with boundary.
method Analyzing geodesics, boundary conditions, and manifold structure.
result All free boundary geodesics have the same length and Morse index.
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 proves the existence and properties of geodesics on convex surfaces.
problem Existence and properties of geodesics on convex surfaces with free boundaries.
method Free boundary curve shortening flow on closed surfaces with strictly convex boundary.
result Existence of two free boundary embedded geodesics and geodesics with Morse Index 1 and 2.
Projective structures are mostly rigid at the boundary but some are not.
problem Boundary rigidity of projective structures.
method Investigation of projective structures on manifolds with boundary.
result Existence of non-rigid projective structures and characterization of them.
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 consider the classification problem and focus on nonlinear methods for classification on manifolds. For multivariate datasets lying on an embedded nonlinear Riemannian manifold within the higher-dimensional ambient space, we aim to acquire a classification boundary for the classes with labels, using the intrinsic me…
We prove surfaces are unknotted with specific properties.
problem Unknottedness of free boundary minimal surfaces and self-shrinkers.
method Introduced concepts of boundary graph and graph at infinity to prove unknottedness.
result Proved surfaces are unknotted with specific properties.
We develop the notion of Brakke flow with free-boundary in a barrier surface. Unlike the classical free-boundary mean curvature flow, the free-boundary Brakke flow must "pop" upon tangential contact with the barrier. We prove a compactness theorem for free-boundary Brakke flows, define a Gaussian monotonicity formula v…
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…
Study boundary structure of gauge fields on AdS spaces.
problem Understanding boundary conditions of gauge fields on asymptotically AdS backgrounds.
method Employing gauge PDE approach to incorporate boundary-defining function.
result Construction of efficient boundary calculus for gauge fields on AdS backgrounds.
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 Dirac-Einstein equations on manifolds with boundary, focusing on constant volume and chiral conditions.
problem Analyzing Dirac-Einstein equations on manifolds with boundary conditions.
method Characterizing bubbling phenomena and classifying ground state bubbles, proving an Aubin-type inequality.
result Proved an Aubin-type inequality and existence result.
Study boundary actions on CAT(0) spaces, proving topological freeness.
problem Understanding boundary actions on CAT(0) spaces.
method Verification of freeness of Myrberg points on boundaries.
result Large class of boundary actions are topologically free.
In the Euclidean unit three-ball, we construct compact, embedded, two-sided free boundary minimal surfaces with connected boundary and prescribed high genus, by a gluing construction tripling the equatorial disc. Aside from the equatorial disc itself, these are the first examples in the three-ball of compact free bound…