arXiv research

The study extends classical results on harmonic functions to Riemannian manifolds with non-tangential boundary limits.

New conditions ensure points can be uniquely represented by combinations of variety elements.

New manifold construction yields Baire-1 functions as cohomotopy groups.

The study examines exceptional sets for radial limits of superharmonic functions on curved manifolds.

Let $R$ be a compact surface and let $Γ$ be a Jordan curve which separates $R$ into two connected components $Σ_1$ and $Σ_2$. A harmonic function $h_1$ on $Σ_1$ of bounded Dirichlet norm has boundary values $H$ in a certain conformally invariant non-tangential sense on $Γ$. We show that if $Γ$ is a quasicircle, then th…

Paper introduces a new Poisson kernel for strongly pseudoconvex domains.

In this paper we study the automorphism group of smoothly bounded convex domains. We show that such a domain is biholomorphic to a "polynomial ellipsoid" (that is, a domain defined by a weighted homogeneous balanced polynomial) if and only if the limit set of the automorphism group intersects at least two closed comple…

Proof confirms Witten's conjecture for a class of 3-spheres.

Let $(φ_t)$ be a holomorphic semigroup of the unit disc (i.e., the flow of a semicomplete holomorphic vector field) without fixed points in the unit disc and let $Ω$ be the starlike at infinity domain image of the Koenigs function of $(φ_t)$. In this paper we completely characterize the type of convergence of the orbit…

Study on the geometry of limit spaces of manifolds with boundary.

Study shows non-symmetric convex sets have full boundary limits.

Study boundary behavior of limit interfaces in Riemannian manifolds without convexity assumptions.

Study limits of curved spaces with boundaries.

The paper characterizes subgroup stability via limit sets on the Morse boundary.

Study ratio-limit boundaries for random walks on hyperbolic groups.

We consider the sharp interface limit of the Allen-Cahn equation with Dirichlet or dynamic boundary conditions and give a varifold characterization of its limit which is formally a mean curvature flow with Dirichlet or dynamic boundary conditions. In order to show the existence of the limit, we apply the phase field me…

Study contractibility of boundaries in convex sets and limit sets of subgroups.

This paper classifies symplectic and Stein fillings of contact 3-manifolds with spinal open book decompositions.

Rare Teichmüller disks converge to small limit sets.

Study G-H limits of surfaces with boundary, focusing on same Euler characteristic.

We consider sequences of open Riemannian manifolds with boundary that have no regularity conditions on the boundary. To define a reasonable notion of a limit of such a sequence, we examine "$δ$ inner regions" which avoid the boundary by a distance $δ$. We prove Gromov-Hausdorff compactness theorems for sequences of the…

The study finds discrete subgroups with full limit sets in higher rank Lie groups.

Study the singular limit of a boundary reaction equation, showing energy concentration and varifold support.

The limit of energies of a sequence of harmonic maps as their annular domains approach the boundary of moduli space depends upon the boundary point approached. The infinite energy case is associated with limits of images containing ruled surfaces. The finite energy case yields a limit of images, under a suitable topolo…

Study inradius collapsed manifolds with lower Ricci curvature bounds, proving properties of their limits.

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…

We study a singular limit problem of the Allen-Cahn equation with Neumann boundary conditions and general initial data of uniformly bounded energy. We prove that the time-parametrized family of limit energy measures is Brakke's mean curvature flow with a generalized right angle condition on the boundary.

For convex domains with $C^{1,ε}$ boundary we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different closed complex faces of the boundary, then the automorphism group has finitely many components and the connected component of the identity is th…

The A-polynomial of a manifold whose boundary consists of a single torus is generalised to an eigenvalue variety of a manifold whose boundary consists of a finite number of tori, and the set of strongly detected boundary curves is determined by Bergman's logarithmic limit set, which describes the exponential behaviour …

Linearized neural networks provide a fast and interpretable way to adapt models to new settings.

Sharp bound on singular set dimension for specific geometric problems.

We study direct limits of embedded Cantor sets and embedded \sier curves. We show that under appropriate conditions on the embeddings, all limits of Cantor spaces give rise to homeomorphic spaces, called $ω$-Cantor spaces, and similarly, all limits of \sier curves give homeomorphic spaces, called to $ω$-\sier curves. W…

Under standard local boundary conditions or certain global APS boundary conditions, we get lower bounds for the eigenvalues of the Dirac operator on compact spin manifolds with boundary. Limiting cases are characterized by the existence of real Killing spinors and the minimality of the boundary.

The paper analyzes thin-shell limits for viscous operators on Riemannian hypersurfaces.

New findings on hyperbolic groups and their boundaries.

In this paper, we give a sharp lower bound for the first eigenvalue of the basic Laplacian acting on basic $1$-forms defined on a compact manifold whose boundary is endowed with a Riemannian flow. The limiting case gives rise to a particular geometry of the flow and the boundary. Namely, the flow is a local product and…

We answer a question of Durham, Hagen, and Sisto, proving that a Teichmüller geodesic ray does not necessarily converge to a unique point in the hierarchically hyperbolic space boundary of Teichmüller space. In fact, we prove that the limit set can be almost anything allowed by the topology.

Thurston's boundary to the universal Teichmüller space $T(\mathbb{H})$ is the set of asymptotic rays to the embedding of $T(\mathbb{H})$ in the space of geodesic currents; the boundary is identified with the projective bounded measured laminations $PML_{bdd}(\mathbb{H})$ of $\mathbb{H}$. We prove that each Teichmüller …

Study gradient flow of phase transitions with fixed contact angle.

We show that the eigenvalues of the intrinsic Dirac operator on the boundary of a Euclidean domain can be obtained as the limits of eigenvalues of Euclidean Dirac operators, either in the domain with a MIT-bag type boundary condition or in the whole space, with a suitably chosen zero order mass term.

In this paper, we present a new method for calculating the limit of early exercise boundary at expiry. We price American style of general derivative using a formula expressed as a sum of the value of European style of derivative and so called American premium. We use the latter expression to calculate an analytic formu…

Suppose $M$ is a complete, embedded minimal surface in $\mathbb{R}^3$ with an infinite number of ends, finite genus and compact boundary. We prove that the simple limit ends of $M$ have properly embedded representatives with compact boundary, genus zero and with constrained geometry. We use this result to show that if …

Study random walks on CAT(0) spaces with contracting elements, proving limit laws.

We study the free boundary Euler equations in two spatial dimensions. We prove that if the boundary is sufficiently regular, then solutions of the free boundary fluid motion converge to solutions of the Euler equations in a fixed domain when the coefficient of surface tension tends to infinity.

We investigate the limit behaviour of sequences of free boundary minimal hypersurfaces with bounded index and volume, by presenting a detailed blow-up analysis near the points where curvature concentration occurs. Thereby, we derive a general quantization identity for the total curvature functional, valid in ambient di…

We construct a Teichmuller geodesic which does not have a limit on the Thurston boundary of the Teichmuller space.

In this paper, we study collapsed manifolds with boundary, where we assume a lower sectional curvature bound, two sides bounds on the second fundamental forms of boundaries and upper diameter bound. Our main concern is the case when inradii of manifolds converge to zero. This is a typical case of collapsing manifolds w…

We study boundary value problems for the Dirac operator on Riemannian Spin$^c$ manifolds of bounded geometry and with noncompact boundary. This generalizes a part of the theory of boundary value problems by C. Bär and W. Ballmann for complete manifolds with closed boundary. As an application, we derive the lower bound …