Extended Vaisman theorem to compact spaces with singularities.
problem Generalizing Vaisman's theorem to spaces with singularities.
method Extended Vaisman's theorem to compact complex spaces with singularities.
result Vaisman's theorem extended to compact spaces with singularities.
In this paper we develop the compactness theorem for λ-surface in R3 with uniform λ, genus, and area growth. This theorem can be viewed as a generalization of Colding-Minicozzi's compactness theorem for self-shrinkers in R3. As an application of this compactness theorem, we prove a rigidity th…
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.
We prove an abstract compactness theorem for a family of generalized Seiberg-Witten equations in dimension three. This result recovers Taubes' compactness theorem for stable flat PSL2(C)-connections as well as the compactness theorem for Seiberg-Witten equations with multiple spinors. Furt…
The paper proves an index theorem for loop spaces of compact manifolds.
problem Defining an index theorem for loop spaces of compact manifolds.
method Formulated and proved an equivariant index theorem for non-compact manifolds with S1-actions, using a ring of formal power series. result Found an appropriate form of the index theorem for loop spaces.
Proves Hamilton's theorem using mean curvature flow.
problem Compactness of pinched hypersurfaces with bounded curvature.
method Mean curvature flow to prove Hamilton's theorem.
result Rigorous proof of Hamilton's theorem.
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.
The paper proves injectivity and vanishing theorems on compact Kahler manifolds.
problem Injectivity and vanishing theorems on compact Kahler manifolds.
method Hodge theory, Bochner-Kodaira-Nakano identity, analytic method, transcendental method, Demailly-Peternell-Schneider equisingular approximation theorem, Hormander L2 estimates.
result The main injectivity theorem implies several Nadel type vanishing theorems.
Proves compactness for timed-metric spaces using new distance and maps.
problem Weak convergence of space-times using timed-Hausdorff distance.
method Uses Gromov's original compactness theorem and introduces addresses.
result Establishes compactness theorem for intrinsic timed-Hausdorff convergence.
Study on symmetric operators on non-compact manifolds, focusing on their index modulo 2.
problem Investigating elliptic operators with a specific symmetry and their index modulo 2.
method Analysis of Callias-type operators on non-compact manifolds, establishing mod 2 versions of index theorems.
result Established mod 2 versions of the Gromov-Lawson relative index theorem, Callias index theorem, and Boutet de Monvel's index theorem for Toeplitz operators.
Vanishing theorem for certain tensor fields on compact Hermitian manifolds.
problem Vanishing theorem for holomorphic tensor fields on compact Hermitian manifolds.
method Inspired by X. Yang and L. Ni-F. Zheng's ideas, the proof uses the definiteness of holomorphic sectional curvature.
result Spaces of certain holomorphic tensor fields are trivial under the definiteness of holomorphic sectional curvature.
Paper proves stability of positive mass theorem for specific types of manifolds.
problem Stability of positive mass theorem for compact graphical manifolds.
method Used Federer--Fleming flat distance and static quasi-local Brown-York energy.
result Proved stability of positive mass theorem for compact (locally) hyperbolic graphical manifolds.
The paper establishes a new sphere theorem for certain types of manifolds.
problem Finding conditions under which compact manifolds are spheres.
method Developed a generalized sphere theorem for manifolds with radial Ricci curvature.
result Established conditions for compact manifolds to be topologically spheres.
Compact theorem for minimal surfaces with lower injectivity radius.
problem Proving compactness of minimal surfaces with lower injectivity radius.
method Variant of Choi--Schoen compactness theorem, focusing on injectivity radius.
result Proved compactness theorem for minimal surfaces.
Compact theorem on Hamiltonian stationary submanifolds in symplectic manifolds.
problem Compactness of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds.
method Proving a compactness theorem with area and extrinsic curvature bounds.
result Uniform bounds on area and total extrinsic curvature lead to compactness of Hamiltonian stationary Lagrangian submanifolds.
The paper extends Bonnet-Myers theorem for manifolds with nonnegative Ricci curvature.
problem Compactness and diameter estimation for manifolds with nonnegative Ricci curvature.
method General curvature conditions for estimating diameter and compactness criteria.
result Established compactness theorems for manifolds with polynomial or exponential Ricci curvature decay.
We prove a compactness theorem for embedded measured hyperbolic Riemann surface laminations in a compact almost complex manifold (X,J). To prove compactness result, we show that there is a suitable topology on the space of measured Riemann surface laminations induced by Levy-Prokhorov metric. As an application of th…
We prove in this paper that, under suitable coinditions on an initial data set, we can obtain Area and Curvature Estimates for simple marginally outer trapped surfaces (or MOTS). Using this estimates, we derive a Compactness Theorem for MOTS. Moreover, the Compactness Theorem will allow us to adapt the recent Degree Th…
Smooth compactness theorem for elasticae, except straight segments.
problem Compactness of elasticae space.
method Smooth compactness theorem proof.
result Smooth stability results for minimizers.
In this note we establish several versions of a compactness theorem for submanifolds. In particular we require only bounds on the second fundamental form and do not assume volume or diameter bounds. As an application we prove a compactness theorem for mean curvature flows and use it to construct smooth blow-up limits a…
The Hodge theorem connects cohomology groups on compact Kähler manifolds.
problem Establishing a relationship between cohomology groups on compact Kähler manifolds.
method Proving the Hodge decomposition theorem on compact d-Kähler manifolds.
result Hodge decomposition theorem on compact d-Kähler manifolds.
The paper computes a residue density for a specific Laplacian on compact manifolds.
problem Computing a noncommutative residue density for a complex Laplacian.
method Explicit computation of the noncommutative residue density associated with equivariant twisted Bismut Laplacian with torsion.
result Proves equivariant twisted Kastler-Kalau-Walze type theorems with torsion on compact manifolds with boundary.
Theorem proves congruence for compact submanifolds in a sphere.
problem Understanding submanifolds in a sphere with specific embedding properties.
method Used a Reilly type formula for space forms.
result Proved a congruence theorem for compact embedded hypersurfaces.
We prove the following theorem for Holomorphic Foliations in compact complex kaehler manifolds: if there is a compact leaf with finite holonomy, then every leaf is compact with finite holonomy. As corollary we reobtain stability theorems for compact foliations in Kaehler manifolds of Edwards-Millett-Sullivan and Hollma…
Vaisman's theorem extended to locally reducible Kähler spaces.
problem Existence of locally conformally Kähler metrics on compact Kähler spaces.
method Extended Vaisman's theorem to locally reducible Kähler spaces.
result Vaisman's theorem holds for compact Kähler spaces that are locally reducible.
We prove a parametrized compactness theorem on manifolds of bounded Ricci curvature, upper bounded diameter and lower bounded injectivity radius.
Absolute index theorem for warped product manifolds.
problem Equivariant index computation for manifolds with warped product structures.
method Warped product structure, Fredholm operator, Atiyah-Segal-Singer index theorem.
result Equivariant relative index theorem for manifolds with warped product structures.
The paper extends the collar theorem to non-compact surfaces using new comparison theorems.
problem Proving the collar theorem for non-compact surfaces.
method Developed new Toponogov-type triangle comparison theorems.
result Eliminated the compactness hypothesis for the collar theorem.
Gradient bounds and Liouville theorems for quasi-linear equations on manifolds with nonnegative Ricci curvature.
problem Establishing bounds and theorems for solutions to quasi-linear elliptic equations on compact manifolds with nonnegative Ricci curvature.
method Gradient bounds, Liouville-type theorems, local splitting theorem, Harnack-type inequality, ABP estimate.
result Gradient bounds and Liouville-type theorems for solutions to quasi-linear equations on compact manifolds with nonnegative Ricci curvature.
The Serre-Swan theorem provides the link between projective modules of finite rank and vector bundles over compact manifolds, and plays a prominent role in non-commutative geometry. Its extension to non-compact manifolds is discussed.
In this paper, we prove some rigidity theorems for compact Bach-flat n-manifold with the positive constant scalar curvature. In particular, our conditions in Theorem 1.4 have the additional properties of being sharp.
Survey of recent Kleinian representation convergence results.
problem Kleinian representation convergence
method Survey and analysis of recent results following Thurston's theorems
result Survey of recent and less recent results on convergence of Kleinian representations
In this paper we use a dynamical approach to prove some new divergence theorems on complete non-compact Riemannian manifolds.
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 study locally compact contractive local groups, that is, locally compact local groups with a contractive pseudo-automorphism. We prove that if such an object is locally connected, then it is locally isomorphic to a Lie group. We also prove a related structure theorem for locally compact contractive local groups whic…
Defines and proves generalized noncommutative residue theorems for specific dimensions.
problem Defining and proving residue theorems for noncommutative geometry.
method Defined generalized noncommutative residue of Dirac operator; proved Kastler-Kalau-Walze type theorems.
result Validated Kastler-Kalau-Walze type theorems for 4D and 6D compact manifolds.
Proves an equivariant version of index theorem for geometric families.
problem Index theorem for geometric families with group action.
method Apply equivariance --> families principle to Clifford module bundles.
result Equivariant version of Bismut's families index theorem.
The paper proves two theorems for modified Novikov operators under conformal perturbations.
problem Proving theorems for modified Novikov operators under conformal perturbations.
method Two Kastler-Kalau-Walze type theorems for conformal perturbations of modified Novikov Operators on 4D and 6D compact manifolds.
result Obtained two Kastler-Kalau-Walze type theorems for conformal perturbations of modified Novikov Operators.
A Heegaard splitting of an open 3-manifold is the partition of the manifold into two non-compact handlebodies which intersect on their common boundary. This paper proves several non-compact analogues of theorems about compact Heegaard splittings. The main theorem is: if N is a compact, connected, orientable 3-manifold …
Study properties of balanced hyperbolic compact complex manifolds.
problem Understanding cohomology and harmonic spaces of balanced hyperbolic manifolds.
method Proved vanishing theorems and Hard Lefschetz-type theorems for balanced hyperbolic compact complex manifolds.
result Non-existence of certain L1 currents on the universal covering space of a balanced hyperbolic manifold. Researchers extend Gamma index theorem to non-compact spacetimes.
problem Establishing an L2-Gamma index for non-compact spacetimes. method Rewriting L2-Gamma index in terms of spectral flow and connecting to geometric expressions. result Extends Bär and Strohmaier's work to non-compact Cauchy hypersurfaces.
We obtain a new differentiable sphere theorem for compact Lagrangian submanifolds in complex Euclidean space and complex projective space.
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.
Proves gap rigidity theorem for Hermitian symmetric spaces.
problem Gap rigidity problems in compact Hermitian symmetric spaces.
method Dual analogy to Mok's noncompact case theorem, theorem on higher dimensional submanifolds.
result Proves gap rigidity theorem for diagonal curves in tube type spaces.
Uhlenbeck's compactness theorem can be used to analyze sequences of connections with anti-self dual curvature on principal SU(2) bundles over oriented 4-dimensional manifolds. The theorems in this paper give an extension of Uhlenbeck's theorem for sequences of solutions of certain SL(2,C) analogs of the anti-self dual …
Compactness theorem for timed-metric spaces established.
problem Compactness of timed-metric spaces and causality.
method Timed-Gromov--Hausdorff distance and intrinsic timed-Hausdorff distance.
result Induces same notion of convergence as intrinsic timed-Hausdorff distance.
A compactness theorem is proved for a family of Kähler surfaces with constant scalar curvature and volume bounded from below, diameter bounded from above, Ricci curvature bounded and the signature bounded from below. Furthermore, a splitting theorem and some rigidity theorems are proved for Einstein-Maxwell systems.
Paper proves embedding theorem for conformally compact manifolds.
problem Embedding conformally compact manifolds into hyperbolic spaces.
method Proves analogous Nash Embedding Theorem for conformally compact manifolds.
result Conformally compact manifolds can be isometrically embedded into hyperbolic spaces.