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.
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.
The paper studies curvature changes on manifolds with boundary.
problem Investigating conformal deformations of curvature on manifolds with boundary.
method Establishing sufficient conditions for positive scalar curvature and mean convex boundary, exploring further deformation scenarios.
result Conditions for conformal deformations to complete metrics with positive scalar curvature and mean convex boundary.
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.
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.
The paper examines stability of Yamabe boundary problem under perturbations.
problem Stability of Yamabe boundary problem under perturbations of mean curvature and scalar curvature.
method Analyzes stability of the Yamabe boundary problem with respect to perturbations of mean curvature and scalar curvature.
result The stability of the Yamabe boundary problem is proven under perturbations from below, but not from above.
Develops slicing method to prove rigidity of scalar curvature on manifolds with boundary.
problem Understanding positive scalar curvature metrics on manifolds with boundary.
method Minimal slicing via capillary hypersurfaces to prove rigidity statements.
result Proves rigidity statement in dimension 4 for specific geometric conditions.
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.
Localized deformation of scalar curvature and mean curvature on manifolds.
problem Deforming scalar curvature and mean curvature on compact manifolds with boundary.
method Proving localized surjection of scalar curvature and mean curvature map, handling non-variational linearized problem.
result Localized deformations of scalar curvature and mean curvature on compact manifolds are possible.
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.
New scalars measure failure of CC metrics to solve singular Yamabe problem.
problem Measuring failure of CC metrics to solve singular Yamabe problem.
method Introducing conformally invariant scalar curvature quantities along conformal infinity.
result CC boundary curvature scalars compute canonical expansion coefficients for singular Yamabe metrics.
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…
Study on ALH manifolds with boundary, showing surjectivity of scalar curvature map and mass rigidity.
problem Characterizing ALH manifolds with boundary and their mass.
method Scalar curvature deformation analysis and mass rigidity study.
result ALH manifolds that minimize mass integrals are characterized.
Generalizes rigidity of scalar curvature for convex domains.
problem Rigidity of scalar curvature for convex domains.
method Harmonic spinors on convex domains with boundary conditions constructed by Brendle.
result Rigidity results on comparison of scalar curvature and scaled mean curvature on the boundary for any convex domain.
We study topological obstructions to the existence of a Riemannian metric on manifolds with boundary such that the scalar curvature is non-negative and the boundary is mean convex. We construct many compact manifolds with boundary which admit no Riemannian metric with non-negative scalar curvature and mean convex bound…
The paper solves scalar curvature problems under conformal deformation for Riemannian manifolds.
problem Solving scalar curvature problems under conformal deformation for Riemannian manifolds.
method Pointwise conformal deformation, Yamabe equation with Dirichlet boundary conditions.
result Positive, smooth solutions to the Yamabe equation with Dirichlet boundary conditions.
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.
Extends index theory results to manifolds with boundaries.
problem Applying index theory to manifolds with boundaries.
method Extends results of Llarull and Goette-Semmelmann.
result Results extended to manifolds with boundaries.
The study proves unique static manifolds with positive scalar curvature and boundary.
problem Characterizing static three-manifolds with boundary and positive scalar curvature.
method Analyzing Ricci curvature bounds and quotient spaces.
result The only orientable quotient of the Nariai static manifold with boundary Nar−1,1(S2) is the only such manifold with connected boundary under certain conditions. We present several problems and results relating the scalar curvatures of manifolds with mean curvatures of their boundaries
Study finds infinitely many non-radial solutions for negative scalar curvature in higher dimensions.
problem Prescribing scalar and boundary mean curvature in a ball with negative scalar curvature.
method Analyzes the existence of infinitely many non-radial positive solutions in dimensions 5 and above.
result First existence result for negative scalar curvature in higher dimensions.
The study proves a rigidity theorem for compact manifolds with boundary.
problem Rigidity of compact manifolds with boundary in low dimensions.
method Dimension reduction argument for mean curvature, extending Schoen-Yau's for scalar curvature.
result Sharp spherical radius rigidity and best NNSC fill-in in terms of mean curvature.
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 doubling conjecture for positive scalar curvature is proven under certain conditions.
problem Determining when a manifold with a specific boundary condition admits positive scalar curvature.
method Surgery techniques for positive scalar and mean curvature, and existence of area-minimizing hypersurfaces.
result The doubling conjecture holds true for manifolds with certain split conditions on fundamental groups.
Study on nonspin manifolds with spin boundary, showing nonconnectedness and nontrivial fundamental group.
problem Understanding spaces of positive scalar curvature metrics on totally nonspin manifolds.
method Analysis of positive scalar curvature metrics on manifolds with spin boundary, using propagation techniques.
result Spaces of positive scalar curvature metrics are not connected and have nontrivial fundamental groups for certain dimensions.
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.
New theorem on 3-manifolds with curvature and convex boundary.
problem Understanding 3-manifolds with specific curvature and boundary properties.
method Analyzes properties of Riemannian 3-manifolds with nonnegative scalar curvature and mean-convex boundary.
result Shows flatness of certain 3-manifolds containing specific geometric objects.
The paper studies scalar curvature and harmonic forms on 3-manifolds with boundaries.
problem Estimating the Thurston norm on 3-manifolds with boundaries.
method Establishing an identity relating average Euler characteristic, scalar curvature, and mean curvature.
result Characterization of the Thurston norm via scalar curvature and harmonic norm for 3-manifolds.
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.
Study on static manifolds with boundary and rigidity of curvature.
problem Understanding the rigidity of scalar curvature and mean curvature on manifolds with boundary.
method Analyzing maps of scalar curvature in the interior and mean curvature on the boundary, discussing geometric properties of static manifolds.
result Classification and rigidity theorems for simple non-generic domains in space forms and Schwarzschild manifold.
The study classifies quasi-Einstein manifolds with constant scalar curvature.
problem Characterizing quasi-Einstein manifolds with specific curvature properties.
method Classification and construction of examples of quasi-Einstein manifolds.
result Complete classification of quasi-Einstein manifolds with constant scalar curvature.
Smooth approximations bound dihedral angles of convex polytopes.
problem Bounding dihedral angles of convex polytopes.
method Approximating polytopes with smooth hypersurfaces and using geometric relations.
result Established lower bounds on dihedral angles.
In this paper, we use localization algebras to study higher rho invariants of closed spin manifolds with positive scalar curvature metrics. The higher rho invariant is a secondary invariant and is closely related to positive scalar curvature problems. The main result of the paper connects the higher index of the Dirac …
Researchers prove spaces of positive scalar curvature metrics are contractible with symmetry.
problem Contractibility of spaces of invariant positive scalar curvature metrics.
method Combining equivariant Morse theory with conformal deformations and local flexibility properties.
result Spaces of invariant positive scalar curvature metrics are contractible.
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.
New method proves rigidity of minimal hypersurfaces in curved 4-manifolds.
problem Proving rigidity of minimal hypersurfaces in curved 4-manifolds.
method Combining nonnegative 2-intermediate Ricci curvature and strict positivity of scalar curvature, extending Chodosh-Li-Stryker method.
result Rigidity of two-sided free boundary stable minimal hypersurfaces in 4-manifolds with bounded geometry and weakly convex boundary.
We prove that any smooth Riemannian manifold of non-negative scalar curvature and with a strictly mean convex and compact boundary component can be (C^2) extended beyond the component to have non-negative scalar curvature and to enjoy anyone of the following three types of (new) boundary: strictly convex, totally geode…
The article proves the existence of a smooth hypersurface with constant scalar curvature and a specified boundary in hyperbolic space.
problem Existence of a smooth complete hypersurface of constant scalar curvature with a prescribed asymptotic boundary in hyperbolic space.
method The problem is reduced to solving a Dirichlet problem for a fully nonlinear elliptic partial differential equation, which is degenerate along the boundary. New techniques are introduced to establish crucial second order a priori estimates for admissible solutions.
result The existence of a smooth complete hypersurface of constant scalar curvature with a prescribed asymptotic boundary at infinity is proven for all possible curvature values.
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…
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…
Study of manifolds with specific curvature properties using capillary surfaces.
problem Obtaining geometric properties of manifolds with nonnegative scalar curvature and strictly mean convex boundary.
method Use of stable capillary surfaces and Urysohn width to study geometric properties.
result Obtained an obstruction to filling 2-manifolds by 3-manifolds.
We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension n≥3. We prove the existence of such conformal metrics in the cases of n=6,7 or the manifold is spin and some other remai…
Estimates scalar curvature without nonnegativity, showing gap phenomenon on manifolds.
problem Estimating scalar curvature without curvature nonnegativity assumption.
method Derive estimates for scalar curvature and mean curvature on manifolds and domains.
result Show that metrics on even dimensional manifolds with nonzero Euler characteristic are ε-gap distance extremal.
The paper solves a problem related to scalar curvature and boundary metrics.
problem Proving the extensibility of boundary metrics to positive scalar curvature metrics.
method Introducing a fill-in invariant and proving relationships with positive mass theorems.
result The positive mass theorem for asymptotically hyperbolic manifolds implies the same for asymptotically flat manifolds.
We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half three sphere, by deforming conformally its standard metric. Using blow up analysis techniques and minimax arguments, we prove some existence and compactness results.
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 first present a warped product manifold with boundary to show the non-uniqueness of the positive constant scalar curvature and positive constant boundary mean curvature equation. Next, we construct a smooth counterexample to show that the compactness of the set of "lower energy" solutions to the above equation fails…
New rigidity theorem on static manifolds with boundary.
problem Static metrics on manifolds with boundary.
method Obata-type rigidity theorem, sufficient geometric conditions.
result Scalar curvature map can be locally surjective at static metrics on manifolds with boundary.