Introduces electrostatic manifolds with boundary for curvature problems.
problem Prescribed scalar curvature problem in manifolds with boundary.
method Generalizes static manifolds with boundary to include electric fields.
result Establishes rigidity theorems for specific manifolds.
Study families of Morse functions for manifolds with boundary.
problem Characterize degeneracies in 1-parameter families of Morse functions.
method List all possible degeneracies in generic 1-parameter families.
result Identified all degeneracies in generic 1-parameter families.
Defines spectral Einstein functional for manifolds with boundary.
problem Calculating the spectral Einstein functional for manifolds with boundaries.
method Defined spectral Einstein functional associated with the Dirac operator and proved a theorem for 4D manifolds.
result Proof of Kastler-Kalau-Walze type theorem for spectral Einstein functional.
The paper defines a new functional and proves related theorems for manifolds with boundary.
problem Defining and proving theorems for manifolds with boundary.
method Defining the spectral Einstein functional and relating it to the noncommutative residue.
result Proof of Dabrowski-Sitarz-Zalecki type theorems for spectral Einstein functional on 4D manifolds with boundary.
The study proves that certain manifolds with boundary cannot have metrics with positive intermediate curvatures.
problem Proving the nonexistence of metrics with positive intermediate curvatures on manifolds with boundary.
method Curvature obstruction theorems for manifolds with boundary.
result Topologically nontrivial compact manifolds with boundary cannot have metrics of positive m-intermediate curvature if the boundary is m-convex. 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.
This paper extends Dabrowski-Sitarz-Zalecki theorems to manifolds with boundary.
problem Generalizing theorems to manifolds with boundary.
method Extending results of Dabrowski etc. to 4D oriented Riemannian manifolds with boundary.
result Proof of Dabrowski-Sitarz-Zalecki type theorems for manifolds with boundary.
Extended equivariant BV formalism to manifolds with boundaries.
problem Handling manifolds with boundaries in equivariant BV formalism.
method Extension of AKSZ theories to manifolds with boundaries.
result Successfully applied to manifolds with boundaries.
Extends K-cowaist inequality to manifolds with boundary.
problem Applying K-cowaist inequality to manifolds with boundary.
method Extended K-cowaist inequality for generalized Dirac operators on manifolds with boundary.
result New applications of K-cowaist inequality to manifolds with boundary.
In this paper, we define lower dimensional volumes of spin manifolds with boundary. We compute the lower dimensional volume Vol6(1,3) for 6-dimensional spin manifolds with boundary and the gravity on boundary is derived by the noncommutative residue associated with Dirac operators.For 6-dimensional manifo…
Proves a theorem for normal distributions on manifolds with boundary.
problem Normal distributions on manifolds with boundary require a new approach to integration.
method Introduces neat integral manifolds with boundary and conditions for integrability.
result Conditions for integrability expressed in terms of adapted collars and integrability on interior and boundary.
Defines spectral Einstein functionals for sub-Dirac operators on manifolds with boundary.
problem Calculating Einstein-like functionals for sub-Dirac operators.
method Introduced spectral Einstein functional for sub-Dirac operators on manifolds with boundary.
result Proved a theorem for spectral Einstein functions on four-dimensional manifolds.
Proves new theorems for manifolds with boundary.
problem No boundary conditions for manifolds.
method Generalizes Dabrowski-Sitarz-Zalecki theorems to manifolds with boundary.
result Proof of new theorems for even and odd dimensional manifolds with boundary.
In this paper, we consider a fully nonlinear problem on manifolds with boundaries of negative admissible curvatures. As a consequence, we conclude the existence of certain types of metrics on the general differential manifolds with boundaries.
Proves another theorem for odd dimensional manifolds with boundary.
problem Spectral Einstein functional on odd dimensional manifolds with boundary.
method Proof of a theorem using the Dirac operator.
result Another general Dabrowski-Sitarz-Zalecki type theorem proved.
The paper establishes a correspondence between normal distributions and neat foliations on manifolds with boundary.
problem Understanding normal distributions on manifolds with boundary.
method Develops a theory analogous to Stefan and Sussmann's for integrable distributions, focusing on neat foliations.
result A one-to-one correspondence between neatly integrable normal distributions and neat foliations by manifolds with boundary.
This manuscript studies manifolds-with-boundary collapsing in the Gromov-Hausdorff topology. The main aim is an understanding of the relationship of the topology and geometry of a limiting sequence of manifolds-with-boundary to that of a limit space, which is presumed to be without geodesic terminals. The main result e…
The paper introduces spectral Einstein functionals for Dirac operators on manifolds with boundary.
problem Analyzing perturbations of Dirac operators on manifolds with boundary.
method Introduction of spectral Einstein functionals and proof of Dabrowski-Sitarz-Zalecki type theorems.
result Proof of theorems associated with spectral Einstein functionals for perturbations of Dirac operators.
We develop Morse theory for manifolds with boundary. Besides standard and expected facts like the handle cancellation theorem and the Morse lemma for manifolds with boundary, we prove that, under a topological assumption, a critical point in the interior of a Morse function can be moved to the boundary, where it splits…
In this paper, we give a brute-force proof of the Kastler-Kalau-Walze type theorem for 7-dimensional manifolds with boundary about Witten deformation, and give a theoritic explaination of the gravitational action for 7 dimensional manifolds with boundary.
In this paper, we define lower dimensional volumes of compact Riemannian manifolds with boundary. For five dimensional spin manifolds with boundary, we prove a Kastler-Kalau-Walze type theorem associated with one-form perturbations of Dirac operators in this case.
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.
In this paper, we extend Su-Zhang's Cheeger-Mueller type theorem for symmetric bilinear torsions to manifolds with boundary in the case that the Riemannian metric and the non-degenerate symmetric bilinear form are of product structure near the boundary. Our result also extends Bruening-Ma's Cheeger-Mueller type theorem…
The study extends a theorem to bundles on manifolds with boundaries.
problem Extending a theorem to bundles on manifolds with boundaries.
method Using recent anomaly results by Brüning, Ma, and Zhang, the theorem is generalized for a general flat bundle with a unimodular restriction to the boundary.
result An analogous statement for a general flat bundle is proven.
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.
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.
In this paper, we define lower dimensional volumes of spin manifolds with boundary. We compute the lower dimensional volume Vol(2,2) for 5-dimensional and 6-dimensional spin manifolds with boundary and we also get the Kastler-Kalau-Walze type theorem in this case.
The paper studies Morse theory on manifolds with boundaries, constructing cellular structures and estimating critical points.
problem Understanding Morse functions on manifolds with boundaries.
method Constructing a cellular structure and analyzing its algebraic properties.
result Estimation of the number of critical points of a Morse function with boundary conditions.
Inspired by the work of Boris Vertman on refined analytic torsion for manifolds with boundary, in this paper we extend the construction of the Cappell-Miller analytic torsion to manifolds with boundary. We also compare it with the refined analytic torsion on manifolds with boundary. As a byproduct of the gluing formula…
The paper studies static manifolds with boundary and their properties.
problem Properties of static manifolds with boundary.
method Theorems relating topology and geometry, isoperimetric inequality, uniqueness theorems.
result Characterization of the round ball in Euclidean 3-space as the only scalar-flat static manifold with mean-convex boundary.
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.
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.
Global and local estimates for a curvature equation on manifolds with boundary.
problem Estimating modified σ2 curvature equation with boundary conditions. method Global and local C2-estimates established. result Global C2-estimates for the modified σ2 curvature equation. Proves two theorems on odd-dimensional manifolds with boundary.
problem Proving theorems on manifolds with boundaries.
method Proof of theorems using mathematical techniques.
result Proved the general Kastler-Kalau-Walze and Dabrowski-Sitarz-Zalecki type theorems.
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 …
Summary of tensor tomography proofs on manifolds with boundaries.
problem Proving injectivity of tensor tomography on compact Riemannian manifolds with boundaries.
method Summarized proofs from previous studies.
result Summary of proofs for s-injectivity.
For non-compact manifolds with boundary we prove that bounded geometry defined by coordinate-free curvature bounds is equivalent to bounded geometry defined using bounds on the metric tensor in geodesic coordinates. We produce a nice atlas with subordinate partition of unity on manifolds with boundary of bounded geomet…
New Dynkin condition for manifolds with boundary yields bi-Lipschitz equivalence and spectral properties.
problem Characterizing smooth Riemannian manifolds with boundary.
method Introducing a Dynkin-type condition and proving its equivalence to a weighted manifold.
result Bi-Lipschitz equivalence and various spectral properties of manifolds with 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.
We observe after Bayle and Rosales that the Levy-Gromov isoperimetric inequality generalizes to convex manifolds with boundary.
Develops a theorem for a 6D manifold with boundary.
problem No specific problem stated; focuses on extending a theorem.
method Extends a Kastler-Kalau-Walze type theorem to 6D almost product Riemannian spin manifolds with boundary.
result Proves a Kastler-Kalau-Walze type theorem for 6D manifolds.
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.
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 this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators on manifolds with boundary. We also consider applications to curvature prescription problems on manifolds with boundary. We relate Dirichlet and Neumann eigenvalues and put the results dev…
Given a smooth compact manifold with boundary, we show that the subcomplex of the deformed de Rham complex consisting of eigenspaces of small eigenvalues of the Witten Laplacian is canonically isomorphic to the Thom-Smale complex constructed by Laudenbach. Our proof is based on Bismut-Lebeau's analytic localization tec…
Paper proves rigidity of 3-manifolds with boundary using modified Hawking mass.
problem Rigidity of 3-manifolds with boundary under specific geometric conditions.
method Area estimates for free boundary strictly stable two-disks, modified Hawking mass analysis.
result 3-manifolds with boundary are locally isometric to half anti-de Sitter-Schwarzschild manifold.
Study submersions with definite folds on manifolds with boundary into Euclidean spaces.
problem Understanding differential-topological properties of manifolds with boundary under submersions with definite folds.
method Analyzing submersions with definite folds on manifolds with boundary into Euclidean spaces, focusing on restrictions to the boundary and using results for m-functions.
result Obtained restrictions on the diffeomorphism types of the source manifolds and studied the diffeomorphism types and Euler characteristics of manifolds admitting such maps.
We answer the question of when the underlying space of an orbifold is a manifold with boundary in several categories.