arXiv research

We study the boundary conditions in the topologically twisted Chern-Simons matter theories with the Lie 3-algebraic structure. We find that the supersymmetric boundary conditions and the gauge invariant boundary conditions can be unified as the complexified gauge invariant boundary conditions which lead to the supergro…

The paper characterizes boundaries in Turaev-Viro TQFTs and Dijkgraaf-Witten theories.

The paper classifies CMC free boundary hypersurfaces in rotational domains.

Study finds topological restrictions for stable free boundary CMC surfaces in negatively curved settings.

The paper explores the geometry and topology of DNN decision boundaries.

We discuss some consequences of the existence of the holomorphic quadratic Hopf differential on a conformally immersed constant mean curvature topological disc with analytic boundary. In particular, we derive a formula for the mean curvature as a weighted average of the normal curvature of the boundary curve, and a con…

The paper proves conditions for smooth convergence of hyperkaehler 4-manifolds with boundary.

The paper studies $\mathfrak{g}$-stability of surfaces with boundary and derives area estimates.

Physicists explain a mathematical theorem about topological insulators.

We show that elliptic complexes of (pseudo)differential operators on smooth compact manifolds with boundary can always be complemented to a Fredholm problem by boundary conditions involving global pseudodifferential projections on the boundary (similarly as the spectral boundary conditions of Atiyah, Patodi and Singer …

The paper explores new phenomena in boundaries of relatively hyperbolic groups.

Revisit Fenn's table theorem from a differential-topological perspective.

Summing over 3-manifolds using TQFT partition functions.

We show that many Lorentzian manifolds of dimension >2 do not admit a spacelike codimension-one foliation, and that almost every manifold of dimension >2 which admits a Lorentzian metric at all admits one which satisfies the dominant energy condition and the timelike convergence condition. These two seemingly unrelated…

This paper clarifies a global structure of Stokes-Dirac structures used for describing interconnected port-Hamiltonian systems defined on manifolds with non-trivial topology under consistent boundary condition.

This paper is devoted to the problem of prescribing the scalar curvature under zero boundary conditions. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results on the standard half sphere.

A guide for solving first-order elliptic boundary value problems.

Estimates the index of the Laplace operator on planar domains with Robin boundary condition.

Proves hyperbolic 3-manifolds have angle structures under certain conditions.

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 measure, in two distinct ways, the extent to which the boundary region of moduli space contributes to the ``simple type'' condition of Donaldson theory. Using a geometric representative of μ(pt), the boundary region of moduli space contributes 6/64 of the homology required for simple type, regardless of the topology…

Study on stellar models' topology and mass using minimal surfaces.

Let M be a possibly noncompact manifold. We prove, generically in the C^k-topology (k=2,...,\infty), that semi-Riemannian metrics of a given index on M do not possess any degenerate geodesics satisfying suitable boundary conditions. This extends a result of Biliotti, Javaloyes and Piccione for geodesics with fixed endp…

Study on free boundary problems in RCD spaces, proving existence and regularity.

In this paper, we study the Seiberg-Witten equations on a compact 3-manifold with boundary. Solutions to these equations are called monopoles. Under some simple topological assumptions, we show that the solution space of all monopoles is a Banach manifold in suitable function space topologies. We then prove that the re…

In this paper we obtain natural boundary conditions for a large class of variational problems with free boundary values. In comparison with the already existing examples, our framework displays complete freedom concerning the topology of $Y$, the manifold of dependent and independent variables underlying a given proble…

Paper proves index theorem for self-adjoint elliptic boundary problems.

New methods assess topological entanglement in periodic systems.

We show that the Gromov boundary of the free product of two infinite hyperbolic groups is uniquely determined up to homeomorphism by the homeomorphism types of the boundaries of its factors. We generalize this result to graphs of hyperbolic groups over finite subgroups. Finally, we give a necessary and sufficient condi…

In this paper, we survey recent results on index defects of elliptic operators on manifolds with boundary. Index defects are similar to the Hirzebruch signature defects in topology, where the defects appear as the correction terms to the signature formula on manifolds with boundary. For some natural classes of elliptic…

Study determines minimal surfaces from boundary data, proving topological and conformal recoverability.

The study of rigidity theorems on 4-manifolds with boundary.

We investigate compactness phenomena involving free boundary minimal hypersurfaces in Riemannian manifolds of dimension less than eight. We provide natural geometric conditions that ensure strong one-sheeted graphical subsequential convergence, discuss the limit behaviour when multi-sheeted convergence happens and deri…

The paper proves rigidity results for compact initial data sets.

Study submersions with definite folds on manifolds with boundary into Euclidean spaces.

New method preserves topology in Hodge decomposition for scalar and vector fields.

Study shows solutions of differential inclusions are homotopy equivalent in $W^{1,p}$-topology.

This survey/expository article covers a variety of topics related to the "topology at infinity" of noncompact manifolds and complexes. In manifold topology and geometric group theory, the most important noncompact spaces are often contractible, so distinguishing one from another requires techniques beyond the standard …

New topology shows Morse boundaries are topologically invariant.

Study identifies topologies of 3D spaces with boundary.

Let $X$ be a proper CAT(0) space and let $G$ be a cocompact group of isometries of $X$ which acts properly discontinuously. Charney and Sultan constructed a quasi-isometry invariant boundary for proper CAT(0) spaces which they called the contracting boundary. The contracting boundary imitates the Gromov boundary for $δ…

The study pinches conditions for constant mean curvature surfaces in convex 3-manifolds.

Compactness proven for CMC surfaces with bounded topology and boundary length.

Constructs minimal surfaces in balls, maximizing eigenvalues.

Given a compact four-dimensional Riemannian manifold $(M, g)$ with boundary, we study the problem of existence of Riemannian metrics on $M$ conformal to $g$ with prescribed $Q$-curvature in the interior $\mathring{M}$ of $M$, and zero $T$-curvature and mean curvature on the boundary $\partial M$ of $M$. This geometric …

Upper bounds for second Robin eigenvalue on Riemannian surfaces.

Researchers found multiple surfaces with same topological and symmetry properties.

Study boundary actions on CAT(0) spaces, proving topological freeness.