arXiv research

Paper solves local well-posedness for Schrödinger flow into sphere with natural boundary conditions.

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…

In this work we study solutions of the prescribed mean curvature equation over a general domain that do not necessarily attain the given boundary data. To such a solution, we can naturally associate a current with support in the closed cylinder above the domain and with boundary given by the prescribed boundary data an…

The paper studies curvature conditions on manifolds with boundary.

Enhances neural network solvers for PDEs with complex boundary conditions.

Novel boundary conditions for Ricci flow to deform compact manifolds.

Solves curve migration problem with elastic flows.

New resurgent analysis reveals dual $q$-series for Chern-Simons theory crossing natural boundaries.

Paper proves convergence of MDL to Einstein-Hilbert with boundary term.

We discuss the problem of prescribing the mean curvature and conformal class as boundary data for Einstein metrics on 3-manifolds, in the context of natural elliptic boundary value problems for Riemannian metrics.

This paper studies several aspects of asymptotically hyperbolic Einstein metrics, mostly on 4-manifolds. We prove boundary regularity (at infinity) for such metrics and establish uniqueness under natural conditions on the boundary data. By examination of explicit black hole metrics, it is shown that neither uniqueness …

We prove that knowing the length of geodesics joining points on the boundary of a two-dimensional, compact, simple Riemannian manifold with boundary, we can determine uniquely the Riemannian metric up to the natural obstruction.

Study stability of Einstein manifolds with boundary.

We present several problems and results relating the scalar curvatures of manifolds with mean curvatures of their boundaries

Symplectic structure found on projective structures on surfaces with boundary.

The theory of geometric structures on a surface with nonempty boundary can be developed by using a decomposition of such a surface into hexagons, in the same way as the theory of geometric structures on a surface without boundary is developed using the decomposition of such a surface into pairs of pants. The basic elem…

The paper solves a flow problem on surfaces with boundary to converge to the hyperbolic metric.

Study identifies obstructions for solving a 4th-order boundary problem.

Paper studies curvature of stable surfaces meeting at a common boundary.

Neural networks optimize stopping boundaries in financial instruments.

In this paper we systematically describe relations between various structure sets which arise naturally for pairs of compact topological manifolds with boundary. Our consideration is based on a deep analogy between the case of a compact manifold with boundary and the case of a closed manifold pair. This approach also g…

In this paper we study the local magnetic ray transform of symmetric tensor fields up to rank two on a Riemannian manifold of dimension $\geq 3$ with boundary. In particular, we consider the magnetic ray transform of the combinations of tensors of different orders due to the nature of magnetic flows. We show that such …

In this paper we prove the propagation of singularities for the wave equation on differential forms with natural (i.e. relative or absolute) boundary conditions on Lorentzian manifolds with corners, which in particular includes a formulation of Maxwell's equations. These results are analogous to those obtained by the a…

We consider natural conformal invariants arising from the Gauss-Bonnet formulas on manifolds with boundary, and study conformal deformation problems associated to them. The key technique we used is to derive boundary C^2 estimates directly from C^0 estimates for fully nonlinear equations. The main result has appeared i…

Boundary value problems for operators of Dirac type arise naturally in connection with the conformal geometry of surfaces immersed in Euclidean 3--space. Recently such boundary value problems have been successfully applied to a variety of problems from computer graphics. Here we investigate under which conditions these…

This paper controls a boundary term in Huisken's formula for entropy.

Study proves Liouville theorem for specific curvature equations with boundary conditions.

We develop a general method of proving the ellipticity of boundary value problems for the stationary vacuum space time, by showing that the stationary vacuum field equations are elliptic subjected to a geometrically natural collection of boundary conditions in the projection formalism. Using this we prove the manifold …

We describe a relation between Atiyah-Patodi-Singer boundary condition and a global elliptic boundary condition which naturally appears in formulating a splitting formula for a spectral flow, when we decompose the manifold into two components. Then we give a variant of the splitting formula with the Hoermander index as…

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…

We introduce Lagrangian mean curvature flow with boundary in Calabi--Yau manifolds by defining a natural mixed Dirichlet-Neumann boundary condition, and prove that under this flow, the Lagrangian condition is preserved. We also study in detail the flow of equivariant Lagrangian discs with boundary on the Lawlor neck an…

The paper studies Ricci flow on manifolds with boundary, proving existence, uniqueness, and boundary conditions preservation.

We give a general criterion for the Dirichlet problem at infinity (DPI) on a Cartan-Hadamard surface to be solvable, which we primarily use to give the best possible upper radial radial curvature bound for solvability of the DPI, but which is also flexible enough to accommodate flats. In particular, any (upper) radial …

The configuration manifold $M$ of a mechanical system consisting of two unconstrained rigid bodies in $\mathbb{R}^n$, $n\geq 1$, is a manifold with boundary (typically with singularities.) A complete description of the system requires boundary conditions that specify how orbits should be continued after collisions. A b…

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 …

For a manifold with nonpositive curvature, the Martin boundary is described by the behavior of normalized Green's functions at infinity. A classical result by Anderson and Schoen states that if the manifold has pinched negative curvature, the geometric boundary is the same as the Martin boundary. In this paper, we stud…

In this paper, we will compute the dimension of the space of spun and ordinary normal surfaces in an ideal triangulation of the interior of a compact 3-manifold with incompressible tori or Klein bottle components. Spun normal surfaces have been described in unpublished work of Thurston. We also define a boundary map fr…

In this paper we show that for Riemannian manifolds with boundary the natural restriction map is a quasifibration between spaces of metrics of positive scalar curvature. We apply this result to study homotopy properties of spaces of such metrics on manifolds with boundary.

We study symplectic Laplacians on compact symplectic manifolds with boundary. These Laplacians are associated with symplectic cohomologies of differential forms and can be of fourth-order. We introduce several natural boundary conditions on differential forms and use them to establish Hodge theory by proving various fo…

The multisymplectic formalism of field theories developed by many mathematicians over the last fifty years is extended in this work to deal with manifolds that have boundaries. In particular, we develop a multisymplectic framework for first order covariant Hamiltonian field theories on manifolds with boundaries. This w…

To every Gromov hyperbolic space X one can associate a space at infinity called the Gromov boundary of X. Gromov showed that quasi-isometries of hyperbolic metric spaces induce homeomorphisms on their boundaries, thus giving rise to a well-defined notion of the boundary of a hyperbolic group. Croke and Kleiner showed t…

Defines timelike ideal boundary for non-positively curved Lorentzian spaces.

We compare some natural triangulations of the Teichmüller space of hyperbolic surfaces with geodesic boundary and of some bordifications. We adapt Scannell-Wolf's proof to show that grafting semi-infinite cylinders at the ends of hyperbolic surfaces with fixed boundary lengths is a homeomorphism. This way, we construct…

Deep neural networks' decision boundaries move closer to natural images during training.

We study the regularity of the solutions of second order boundary value problems on manifolds with boundary and bounded geometry. We first show that the regularity property of a given boundary value problem $(P, C)$ is equivalent to the uniform regularity of the natural family $(P_x, C_x)$ of associated boundary value …

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

Study isotopy and concordance for metrics on 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…