arXiv research

Boundary of fiber convex domains is a cohomological sphere.

The paper proves estimates for solutions to nonlinear equations on manifolds with boundary.

Boundary Dehn twists become trivial after abelianization.

Generalizes Poincaré-Hopf Theorem for piecewise smooth boundaries.

Smooth solutions up to evolving free boundaries for degenerate equations.

The aim of this note is to study the measure-valued Ricci tensor on smooth metric measure space with boundary, which is a generalization of Bakry-Emery's modified Ricci tensor on weighted Riemannian manifold. As an application, we offer a new approach to study curvature-dimension condition of smooth metric measure spac…

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

The paper proves gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.

Following a survey of the abstract boundary definition of Scott and Szekeres, a rigidity result is proved for the smooth case, showing that the topological structure of the regular part of this boundary in invariantly defined.

Let $N$ be a Riemannian manifold and consider a stationary union of three or more $C^{1,μ}$ hypersurfaces-with-boundary $M_k$ in $N$ with a common boundary $Γ$. We show that if $N$ is smooth, then $Γ$ is smooth and each $M_k$ is smooth up to $Γ$ (real analytic in the case $N$ is real analytic). Consequently we strength…

By means of a general gluing and conformal-deformation construction, we prove that any smooth, metrically complete Riemannian manifold with smooth boundary can be realized as a closed domain into a smooth, geodesically complete Riemannan manifold without boundary. Applications to Sobolev spaces, Nash embedding and loca…

Proves rigidity for specific initial data sets under the dominant energy condition.

Computes mapping class groups of 4-manifolds with boundary.

Smooth solutions found for hydrodynamic equations.

Study improves boundary smoothness for area-minimizing currents with complex boundaries.

Let $M_1$ and $M_2$ be two $n$-dimensional smooth manifolds with boundary. Suppose we glue $M_1$ and $M_2$ along some boundary components (which are, therefore, diffeomorphic). Call the result $N.$ If we have a group $G$ acting continuously on $M_1,$ and also acting continuously on $M_2,$ such that the actions are comp…

We prove weak and strong maximum principles, including a Hopf lemma, for smooth subsolutions to equations defined by linear, second-order, partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the smooth subsolutions along this boundary…

Study constraints on diffeomorphisms and homeomorphisms of 4-manifolds with boundary.

Study solves Yamabe problems on metric measure spaces with or without boundary.

Classifies $4$-manifolds with specific properties and fundamental group.

We show that the travel time difference functions, measured on the boundary, determine a compact Riemannian manifold with smooth boundary up to Riemannian isometry, if boundary satisfies a certain visibility condition. This corresponds with the inverse microseismicity problem. The novelty of our paper is a new type of …

In this paper, we study the regularity of asymptotically hyperbolic metrics with Einstein condition near boundary and Weyl curvature smooth enough in arbitrary dimension. Following Michael Anderson's method, we show that $C^{m,α}$ conformally compact Riemannian metrics with Einstein equation vanishing to finite order n…

On any given compact (n+1)-manifold M with non-empty boundary, it is proved that the moduli space of Einstein metrics on M is a smooth, infinite dimensional Banach manifold under a mild condition on the fundamental group. Thus, the Einstein moduli space is unobstructed. The Dirichlet and Neumann boundary maps to data o…

Study on special Lagrangian curvature potential equation, proving existence and uniqueness of smooth solutions.

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

Generic smooth boundaries for isoperimetric regions in 8D manifolds.

We show how to construct absolutely exotic smooth structures on compact 4-manifolds with boundary, including contractible manifolds. In particular, we prove that any compact smooth 4-manifold W with boundary that admits a relatively exotic structure contains a pair of codimension-zero submanifolds homotopy equivalent t…

Study on elasticity with mixed boundary conditions, proving spectral asymptotics.

Given a compact four dimensional smooth Riemannian manifold $(M,g)$ with smooth boundary, we consider the evolution equation by $Q$-curvature in the interior keeping the $T$-curvature and the mean curvature to be zero and the evolution equation by $T$-curvature at the boundary with the condition that the $Q$-curvature …

Eigenvalue estimate for the Dirac-Witten operator is given on bounded domains (with smooth boundary) of spacelike hypersurfaces satisfying the dominant energy condition, under four natural boundary conditions (MIT, APS, modified APS, and chiral conditions). This result is a generalisation of Friedrich's inequality for …

Study detects boundaries in unlabeled noisy images without labels.

Classifies local boundary conditions for Dirac-type operators on manifolds.

We prove that any two smooth h-cobordant simply-connected 4-manifolds can be obtained by taking two manifolds with boundary, one of which is contractible, and gluing them along the boundary via two different attaching maps.

On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that the Atiyah-Singer Dirac operator $\mathrm{D}_{\mathcal B}$ in $\mathrm{L}^{2}$ depends Riesz continuously on $\mathrm{L}^{\infty}$ perturbations of local boundary conditions ${\mathcal B}$. The Lipschitz bound for the map ${…

Let $M$ be a Riemannian manifold of dimension $n+1$ with smooth boundary and $p\in \partial M$. We prove that there exists a smooth foliation around $p$ whose leaves are submanifolds of dimension $n$, constant mean curvature and its arrive perpendicular to the boundary of M, provided that $p$ is a nondegenerate critica…

Let $N$ be a smooth $(n+l)$-dimensional Riemannian manifold. We show that if $V$ is an area-stationary union of three or more $C^{1,μ}$ $n$-dimensional submanifolds-with-boundary $M_k \subset N$ with a common boundary $Γ$, then $Γ$ is smooth and each $M_k$ is smooth up to $Γ$ (real-analytic in the case $N$ is real-anal…

Theory for capillary surfaces in 3-manifolds with smooth boundary.

We construct Peano curves $γ: [0,\infty) \to \mathbb{R}^2$ whose "footprints" $γ([0,t])$, $t>0$, have $C^\infty$ boundaries and are tangent to a common continuous line field on the punctured plane $\mathbb{R}^2 \setminus \{γ(0)\}$. Moreover, these boundaries can be taken $C^\infty$-close to any prescribed smooth family…

In terms of Turaev's shadows, we provide a sufficient condition for a compact, smooth, acyclic 4-manifold with boundary the 3-sphere to be diffeomorphic to the standard 4-ball. As a consequence, we prove that if a compact, smooth, acyclic 4-manifold with boundary the 3-sphere has shadow-complexity at most 2, then it is…

This paper shows that the time $t$ map of the averaged Euler equations, with Dirichlet, Neumann, and mixed boundary conditions is canonical relative to a Lie-Poisson bracket constructed via a non-smooth reduction for the corresponding diffeomorphism groups. It is also shown that the geodesic spray for Neumann and mixed…

In this paper we find strictly locally convex hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed curvature and boundary. The main result is that if the given data admits a strictly locally convex radial graph as a subsolution, we can find a radial graph realizing the prescribed curvature and boundary. As an applicatio…

Study on controllability and groups of manifolds with boundaries.

We establish the regularity theory for certain critical elliptic systems with an anti-symmetric structure under inhomogeneous Neumann and Dirichlet boundary constraints. As applications, we prove full regularity and smooth estimates at the free boundary for weakly Dirac-harmonic maps from spin Riemann surfaces. Our met…

Algebras of smooth functions help reconstruct bulk topological types.

We investigate the functional determinant of the laplacian on piece-wise flat two-dimensional surfaces, with conical singularities in the interior and/or corners on the boundary. Our results extend earlier investigations of the determinants on smooth surfaces with smooth boundaries. The differences to the smooth case a…

Corners can be identified by a drum's sound spectrum.

Proof shows smooth minimal hypersurfaces for perimeter-minimizing sets in low-dimensional Riemannian manifolds.

Smooth approximations bound dihedral angles of convex polytopes.