arXiv research

Clarifies global structure of Stokes-Dirac structures on manifolds.

Proves well-posedness for Einstein equations with specific boundary conditions.

Enhances deep networks' robustness against adversarial attacks.

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

The critical catenoid is uniquely determined by certain symmetries of its boundary.

Study on well-posedness of vacuum Einstein equations with specific boundary conditions.

We consider Landau-Ginzburg (LG) models with boundary conditions preserving A-type N=2 supersymmetry. We show the equivalence of a linear class of boundary conditions in the LG model to a particular class of boundary states in the corresponding CFT by an explicit computation of the open-string Witten index in the LG mo…

In the previous work ([14]) we introduced the well-posed boundary conditions ${\mathcal P}_{-, {\mathcal L}_{0}}$ and ${\mathcal P}_{+, {\mathcal L}_{1}}$ for the odd signature operator to define the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the gluing formula of the refined…

The odd signature operator is a Dirac operator which acts on the space of differential forms of all degrees and whose square is the usual Laplacian. We extend the result of [15] to prove the gluing formula of the zeta-determinants of Laplacians acting on differential forms of all degrees with respect to the boundary co…

Upper bound found for Steklov eigenvalue of a surface of revolution.

We develop a Kobayashi-Hitchin correspondence for the extended Bogomolny equations, i.e., the dimensionally reduced Kapustin-Witten equations, on the product of a compact Riemann surface $Σ$ with ${\mathbb R}^+_y$, with generalized Nahm pole boundary conditions at $y=0$. The correspondence is between solutions of these…

The paper characterizes a helicoid in a cylinder with minimal area and unique boundary conditions.

Study on static manifolds with boundary and rigidity of curvature.

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…

Deep learning method solves American options with free boundary using Landau transformation.

Localized deformation of scalar curvature and mean curvature on manifolds.

A unique hyperbolic metric is found for each spherical cone-metric on the boundary of a hyperbolizable 3-manifold.

Proposes a framework to reconcile policy learning and profit maximization in CATE estimation.

Let $X$ be a compact manifold with boundary. Suppose that the boundary is fibred, $φ:\pa X\longrightarrow Y,$ and let $x\in\CI(X)$ be a boundary defining function. This data fixes the space of `fibred cusp' vector fields, consisting of those vector fields $V$ on $X$ satisfying $Vx=O(x^2)$ and which are tangent to the f…

We find a new obstruction for a real Einstein 4-orbifold with an A1-singularity to be a limit of smooth Einstein 4-manifolds. The obstruction is a curvature condition at the singular point. For asymptotically hyperbolic metrics, with boundary at infinity a conformal metric, we prove that if the obstruction vanishes, on…

EPGP surrogate outperforms finite elements in solving wave equations.

Sharp generalization of boundary regularity for area minimizing currents with arbitrary multiplicity.

The refined analytic torsion on compact Riemannian manifolds with boundary has been discussed by B. Vertman and the authors, but these two constructions are completely different. Vertman used a double of de Rham complex consisting of the minimal and maximal closed extensions of a flat connection and the authors used we…

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…

It is shown that the non-homogeneous Dirichlet and Neuman problems for the $2^{nd}$-order Seiberg-Witten equation admit a regular solution once the $\mathcal{H}$-condition (described in the article) is satisfied. The approach consist in applying the elliptic techniques to the variational setting of the Seiberg-Witten e…

New method uses broken scattering to uniquely identify Finsler manifolds.

Narasimhan and Ramadas showed that the restricted holonomy group of the Coulomb connection is dense in the connected component of the identity of the gauge group when one considers the product principal bundle $S^3 \times SU(2)\to S^3$. Instead of a base manifold S^3, we consider here a base manifold of dimension $n\ge…

For a compact surface $X_0$, Thurston introduced a compactification of its Teichmüller space $\mathcal T(X_0)$ by completing it with a boundary $\mathcal{PML}(X_0)$ consisting of projective measured geodesic laminations. We introduce a similar bordification for the Teichmüller space $\mathcal T(X_0)$ of a noncompact Ri…

We discuss the constant $σ_{2}$ problem for conic 4-spheres. Based on earlier works of Chang-Han-Yang and Han-Li-Teixeira, we are able to find a necessary condition for the existence problem. In particular, when the condition is sharp, we have the uniqueness result similar to that of Troyanov in dimension 2. It indicat…

Establishes a version of Bartnik's conjecture for Lorentzian length spaces.

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 …

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

Proof of local well-posedness for a specific boundary condition in general relativity.

Study boundary value problems for elliptic operators on manifolds.

Under two boundary conditions, the generalized Atiyah-Patodi-Singer boundary condition and the modified generalized -Atiyah-Patodi-Singer boundary condition, we get the lower bounds for the eigenvalues of the fundamental Dirac operator on compact spin manifolds with nonempty boundary.

A new machine learning method calculates failure probability efficiently and accurately.

We prove that the topology, smooth structure, and metric of a compact Lorentzian manifold with boundary is uniquely determined by data at the boundary. The data consists of the lengths and directions of future-directed once-broken geodesics connecting points on the boundary, which are first timelike and then lightlike.…

New formula for torsion function in 3-manifolds with torus boundaries.

Optimizes maps with controlled distortion for geometric tasks.

New boundary conditions solve Cauchy problem for Dirac operators on spacetimes.

We introduce new boundary conditions for differential forms on symplectic manifolds with boundary. These boundary conditions, dependent on the symplectic structure, allows us to write down elliptic boundary value problems for both second-order and fourth-order symplectic Laplacians and establish Hodge theories for the …

In this paper, we study the Dirichlet problem of the geodesic equation in the space of Kähler cone metrics $\mathcal H_\b$; that is equivalent to a homogeneous complex Monge-Ampère equation whose boundary values consist of Kähler metrics with cone singularities. Our approach concerns the generalization of the space def…

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

We construct maximal hypersurfaces with a Neumann boundary condition in Minkowski space via mean curvature flow. In doing this we give general conditions for long time existence of the flow with boundary conditions with assumptions on the curvature of a the Lorentz boundary manifold.

Novel boundary conditions for Ricci flow to deform compact manifolds.

Study sharp interface limit of Allen-Cahn equation to characterize mean curvature flow with boundary conditions.

PINN-FEM combines PINNs and FEM for accurate Dirichlet boundary condition enforcement.

This paper establishes the geometric structure of the lines of principal curvature of a hypersurface immersed in ${\mathbb R}^4$ in a neighborhood of the set $\mathcal{S}$ of its principal curvature singularities, consisting of the points at which atF least two principal curvatures are equal. Under generic conditions d…