arXiv research

Develops a method to deform metrics on manifolds with non-compact boundaries.

Paper solves Minkowski problem for non-compact convex sets with asymptotic boundary conditions.

Positive mass theorem for non-smooth metrics on flat manifolds with corners.

Study of elliptic boundary value problems on non-compact manifolds.

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 …

We study the Cauchy data spaces of the strongly Callias-type operators using maximal domain on manifolds with non-compact boundary, with the aim of understanding the Atiyah-Patodi-Singer index and elliptic boundary value problems.

Study eta invariant on non-compact manifolds with positive scalar curvature.

In this paper, an equality between the Hochs-Mathai type index and the Atiyah-Patodi-Singer type index is established when the manifold and the group action are both non-compact, which generalizes a result of Ma and Zhang for compact group actions. As a technical preparation, a problem concerning the Fredholm property …

New local method solves Yamabe problems on compact and non-compact manifolds.

We prove a positive mass theorem for $n$-dimensional asymptotically flat manifolds with a non-compact boundary if either $3\leq n\leq 7$ or if $n\geq 3$ and the manifold is spin. This settles, for this class of manifolds, a question posed in a recent paper by the first author in connection with the long-term behavior o…

Paper calculates homotopy types of non-compact surfaces using groupoids.

In this paper we consider non-compact non-flat simply connected harmonic manifolds. In particular, we show that the Martin boundary and Busemann boundary coincide for such manifolds. For any finite volume quotient we show that (up to scaling) there is a unique Patterson-Sullivan measure and this measure coincides with …

The paper proves a positive mass theorem for non-compact static domains in hyperbolic space.

We prove regularity for a class of boundary value problems for first order elliptic systems, with boundary conditions determined by spectral decompositions, under coefficient differentiability conditions weaker than previously known. We establish Fredholm properties for Dirac-type equations with these boundary conditio…

Let $Z$ be a non-compact two-dimensional manifold and $Δ$ be a one-dimensional foliation of $Z$ such that $\partial Z$ consists of leaves of $Δ$ and each leaf of $Δ$ is a non-compact closed subset of $Z$. We obtain a characterization of a subclass of such foliated surfaces $(Z,Δ)$ glued from open strips $\mathbb{R}\tim…

We study Hawking mass and the Huisken's isoperimetric mass evaluated on surfaces with boundary. The convergence to an ADM mass defined on asymptotically flat manifold with a non-compact boundary are proved.

In this article, we prove the Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary whose asymptotic region is modelled on a half-space. Such spaces were initially considered by Almaraz, Barbosa and de Lima in 2014. In order to prove the inequality, we develop a new approximation sch…

We consider a class of globally hyperbolic space-times with "expanding singularities". Under suitable assumptions we show that no $C^0$-extensions across a compact boundary exist, while the boundary must be null wherever differentiable (which is almost everywhere) in the non-compact case.

The study extends conserved quantities theory to non-compact boundary initial data sets.

We consider a hyperbolic Dirac-type operator with growing potential on a a spatially non-compact globally hyperbolic manifold. We show that the Atiyah-Patodi-Singer boundary value problem for such operator is Fredholm and obtain a formula for this index in terms of the local integrals and the relative eta-invariant int…

Extends Atiyah-Singer Dirac operator study to non-compact spacetimes.

We study non-compact surfaces obtained by gluing strips $\mathbb{R}\times(-1,1)$ with at most countably many boundary intervals along some these intervals. Every such strip possesses a foliation by parallel lines, which gives a foliation on the resulting surface. It is proved that the identity path component of the gro…

Non-compact convex sets in hyperbolic 3-space are rigid under isometries.

In this work we prove the existence of embedded closed minimal hypersurfaces in non-compact manifolds containing a bounded open subset with smooth and strictly mean-concave boundary and a natural behavior on the geometry at infinity. For doing this, we develop a modified min-max theory for the area functional following…

In this note, we extend the Bridgeman-Kahn identity to all finite-volume orientable hyperbolic $n$-manifolds with totally geodesic boundary. In the compact case, Bridgeman and Kahn are able to express the manifold's volume as the sum of a function over only the orthospectrum. For manifolds with non-compact boundary, ou…

Introduce new boundary mass for asymptotically flat half-manifolds

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…

We show that canonical Carnot-Caratheodory spherical and horospherical metrics, which are defined on the boundary at infinity of every rank one symmetric space of non-compact type, are visual, i.e., they are bilipschitz equivalent with universal bilipschitz constants to the inverse exponent of Gromov products based in …

Given any connected, open 3-manifold $U$ having finitely many ends, a non-compact 3-manifold $M$ is constructed having the following properties: the interior of $M$ is homeomorphic to $U$; the boundary of $M$ is the disjoint union of finitely many planes; $M$ is not almost compact; $M$ is eventually end-irreducible; th…

We prove capacity inequalities involving the total mean curvature of hypersurfaces with boundary in convex cones and the mass of asymptotically flat manifolds with non-compact boundary. We then give the analogous of Pölia-Szegö, Alexandrov-Fenchel and Penrose type inequalities in this setting. Among the techniques used…

Develops sublinear Morse theory in symmetric spaces.

The paper studies how certain surfaces evolve in space-time.

Constructs real algebraic functions with both compact and non-compact preimages.

Proves positive mass theorem for 3-manifolds with a boundary.

New equations simplify gauge-theoretic Khovanov homology solutions.

We consider a complete Riemannian manifold M whose boundary is a disjoint union of finitely many complete connected Riemannian manifolds. We compute the index of a local boundary value problem for a strongly Callias-type operator on M. Our result extends an index theorem of D. Freed to non-compact manifolds, thus provi…

Study on free boundary minimal hypersurfaces in Schwarzschild space, proving zero Morse index for certain hypersurfaces.

The purpose of this article is to prove existence of mass minimizing integral currents with prescribed possibly non-compact boundary in all dual Banach spaces and furthermore in certain spaces without linear structure, such as injective metric spaces and Hadamard spaces. We furthermore prove a weak$^*$-compactness theo…

Study on metrics on manifolds with specific curvature properties.

We investigate the Berezin integral of non-compactly supported quantities. In the framework of supermanifolds with corners, we give a general, explicit and coordinate-free repesentation of the boundary terms introduced by an arbitrary change of variables. As a corollary, a general Stokes's theorem is derived - here, th…

Minimal graphs over non-compact domains in 3-manifolds solved with estimates and uniqueness results.

In this article a class of closed convex sets in the Euclidean $n$-space which are the convex hull of their profiles is described. Thus a generalization of Krein-Milman theorem\cite{Lay:1982} to a class of closed non-compact convex sets is obtained. Sufficient and necessary conditions for convexity, affinity and starsh…

We show that if an asymptotically flat manifold with horizon boundary admits a global static potential, then the static potential must be zero on the boundary. We also show that if an asymptotically flat manifold with horizon boundary admits an unbounded static potential in the exterior region, then the manifold must c…

Let $X$ be a connected non-compact $2$-dimensional manifold possibly with boundary and $Δ$ be a foliation on $X$ such that each leaf $ω\inΔ$ is homeomorphic to $\mathbb{R}$ and has a trivially foliated neighborhood. Such foliations on the plane were studied by W. Kaplan who also gave their topological classification. H…

The paper studies the connectedness of a graph's boundary for surfaces.

Calderón projector extended to fibred cusp operators.

Proves spacetime positive mass theorem for spin initial data sets with arbitrary ends.

Uniform proof reconstructs spaces using cross ratio on boundary.