Complex domains covering manifolds are biholomorphic to balls.
problem Covering compact manifolds by bounded domains with smooth boundaries.
method Using biholomorphic mappings and properties of C1,1 boundaries. result Bounded domains with smooth boundaries covering compact manifolds are biholomorphic to the unit ball.
New geometric conditions ensure compactness of ∂ˉ-Neumann problem.
problem Compactness of ∂ˉ-Neumann operator on specific domains. method Introduced new geometric conditions for a class of domains, proving compactness equivalence to boundary properties.
result Compactness of ∂ˉ-Neumann operator equivalent to boundary lack of analytic varieties. Ideal Liouville domains simplify symplectic structures.
problem Inconvenience from Liouville forms and non-compactness.
method Defining compact manifolds with boundary and symplectic forms.
result Ideal Liouville domains suppress awkward aspects.
Magnitude of Euclidean domains predicts Willmore energy in odd dimensions.
problem Magnitude function of compact domains in odd dimensions.
method Asymptotic expansion of magnitude function at infinity.
result Magnitude function determines Willmore energy of boundary in odd dimensions.
Generalizes twistor lines for complex tori, introducing new non-compact curves.
problem Understanding the structure of complex tori through twistor lines.
method Introducing and studying two new types of non-compact analytic curves in the period domain of complex tori.
result Analytic properties of compactifications of curves, preservation of cohomology classes, and twistor path connectivity.
The paper shows neural networks can approximate functions over non-compact domains with non-polynomial activation.
problem Approximating functions over non-compact domains using neural networks.
method Using single-hidden-layer feedforward neural networks with non-polynomial activation functions over non-compact subsets of Euclidean spaces.
result Neural networks can approximate functions in weighted Ck-spaces and weighted Sobolev spaces over unbounded domains. Fisher loss improves deep domain adaptation by learning discriminative within-class compact and between-class separable representations.
problem Improving deep domain adaptation performance by learning discriminative representations.
method Proposes a Fisher loss to learn discriminative representations that are within-class compact and between-class separable.
result Noticeable improvements in deep domain adaptation performance, e.g., 6.67% absolute improvement in mean accuracy on the Office-Home dataset.
Study characterizes totally geodesic submanifolds in quotient spaces.
problem Characterizing totally geodesic submanifolds in quotient spaces.
method Characterization through totally geodesic submanifolds and holomorphic tangent sequence splitting.
result Characterization of totally geodesic submanifolds in quotient spaces.
Study on unique minimal hypersurfaces in rotational domains.
problem Existence of compact free-boundary minimal hypersurfaces in rotational domains.
method Integral identity for compact free-boundary minimal hypersurfaces, applied to rotational domains.
result Existence of minimal hypersurfaces in rotational domains without topological restrictions.
Study on Lagrangian phase operator solutions in compact domains.
problem Existence of solutions to the Dirichlet problem for the Lagrangian phase operator.
method Analysis of subsolutions and boundary conditions.
result Existence of solutions depends on the existence of subsolutions.
We study eigenvalues of polyharmonic operators on compact Riemannian manifolds with boundary (possibly empty). In particular, we prove a universal inequality for the eigenvalues of the polyharmonic operators on compact domains in a Euclidean space. This inequality controls the kth eigenvalue by the lower eigenvalues,…
Study approximates kinetic VFP in bounded domains using weak compactness.
problem Approximating Vlasov-Fokker-Planck in bounded domains with boundary conditions.
method Weak compactness in weighted Hilbert space, construction of test functions.
result Derives diffusion approximation for Vlasov-Fokker-Planck in bounded domains.
Consider a strictly convex bounded regular domain C of R3. For any arbitrary finite topological type we find a compact Riemann surface M, an open domain M⊂M with the fixed topological type, and a conformal complete proper minimal immersion X:M→C which can be extended to a conti…
In this article we prove a reverse Hölder inequality for the fundamental eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with lower Ricci curvature bounds. We also prove an isoperimetric inequality for the torsional ridigity of such domains.
The paper finds symplectic compactifications of coadjoint orbits.
problem Understanding symplectic structures on coadjoint orbits.
method Defined real analytic symplectomorphisms on subsets of coadjoint orbits.
result Coadjoint orbits of compact Lie algebras are symplectic compactifications of domains of cotangent bundles.
The study of Kähler metrics on domains restricts their boundary geometry.
problem Understanding the geometry of domains with negatively pinched Kähler metrics.
method Analyzing the existence and properties of negatively pinched Kähler metrics on domains.
result The boundary of a convex domain without complex subvarieties of positive domain if it admits a complete Kähler metric with pinched negative holomorphic bisectional curvature.
NFM models time-series data directly in the Fourier domain, achieving state-of-the-art performance.
problem Traditional time-series analysis focuses on the time domain, limiting flexibility.
method NFM models time-series data in the Fourier domain, using frequency extrapolation and interpolation.
result NFM achieves state-of-the-art performance on various time-series tasks.
The paper proves a flat torus theorem for certain groups acting on convex domains.
problem Establishing a flat torus theorem for specific groups acting on convex domains.
method Analyzing discrete groups in mPGLd(R) acting convex co-compactly on a properly convex domain. result An analogue of the flat torus theorem for mCAT(0) spaces is proven for these groups. In this paper, we prove that every conformal minimal immersion of a compact bordered Riemann surface M into a minimally convex domain D⊂R3 can be approximated, uniformly on compacts in M˚=M∖bM, by proper complete conformal minimal immersions M˚→D. We also obtain a …
The paper defines conditions for groups acting on convex domains to be relatively hyperbolic.
problem Understanding conditions for groups acting on convex domains to be relatively hyperbolic.
method Analyzing the geometry of the convex domain to determine relative hyperbolicity.
result Established necessary and sufficient conditions for groups to be relatively hyperbolic.
Paper proves uniqueness of minimal hypersurfaces in specific domains.
problem Proving uniqueness of minimal hypersurfaces in constrained domains.
method Analyzing flat and compact free boundary minimal hypersurfaces in Euclidean balls and annular domains.
result Uniqueness of minimal hypersurfaces in unit Euclidean ball and annular domains.
New extrinsic lower bounds are given for the classical Dirac operator on the boundary of a compact domain of a spin manifold. The main tool is to solve some boundary problems for the Dirac operator of the domain under boundary conditions of Atiyah-Patodi-Singer type. Spinorial techniques are used to give simple proofs …
In this article, we study convex affine domains which can cover a compact affine manifold. For this purpose, we first show that every strictly convex quasi-homogeneous projective domain has at least C1 boundary and it is an ellipsoid if its boundary is twice differentiable. And then we show that an n-dimensional par…
Inspired by the work of Z. Lu and G. Tian [21] in the compact setting, in this paper we address the problem of studying the Szegö kernel of the disk bundle over a noncompact Kähler manifold. In particular we compute the Szegö kernel of the disk bundle over a Cartan-Hartogs domain based on a bounded symmetric domain. Th…
Paper proves geodesic ball maximizes second Robin eigenvalue in non-compact symmetric spaces.
problem Maximizing the second Robin eigenvalue in non-compact rank-1 symmetric spaces.
method Quantitative spectral inequality for the second Robin eigenvalue.
result Geodesic ball maximizes the second Robin eigenvalue among domains of the same volume.
Extends polydisk theorem to Hartogs domains over symmetric domains.
problem Rigidity phenomena in Riemannian manifolds.
method Extension of polydisk theorem to Hartogs domains over arbitrary symmetric domains.
result Dual of a Hartogs domain over a bounded symmetric domain admits no totally geodesic immersion into any compact Riemannian manifold.
Paper proves eigenvalue inequality for Hopf-symmetric domains.
problem Eigenvalue inequality for Hopf-symmetric domains in non-compact symmetric spaces.
method Used geometric and spectral analysis on non-compact rank one symmetric spaces.
result Eigenvalue inequality for bounded Hopf-symmetric domains in non-compact symmetric spaces.
In this paper, we establish universal inequalities for eigenvalues of the clamped plate problem on compact submanifolds of Euclidean spaces, of spheres and of real, complex and quaternionic projective spaces. We also prove similar results for the biharmonic operator on domains of Riemannian manifolds admitting spherica…
The paper proves a positive mass theorem for non-compact static domains in hyperbolic space.
problem Proving a positive mass theorem for non-compact static domains in hyperbolic space.
method Formulating and proving a positive mass theorem under natural dominant energy conditions, using elliptic boundary conditions on spinors.
result Retrieve a sharper version of a recent result by Souam about the rigidity of non-compact static domains.
Estimates the index of the Laplace operator on planar domains with Robin boundary condition.
problem Index estimates for planar domains with Robin boundary condition
method Combines conformal and spectral techniques with topology of the domain.
result Lower bounds for the index in terms of the number of boundary components.
The paper proves conditions for convex domains to be strongly pseudoconvex.
problem Conditions for convex domains to be strongly pseudoconvex.
method Establishes gap theorem for complex geometry of convex domains.
result Conditions for convex domains to be strongly pseudoconvex.
We prove that given any compact Riemannian 3-manifold with boundary M, there exists a smooth properly embedded one-manifold G, included in M, each of whose components is a simple closed curve and such that the domain D=Int(M)-G does not admit any properly immersed open surfaces with at least one annular end, bounded me…
Minimum width for ReLU networks on compact domain is exactly max{d_x, d_y, 2}
problem Characterizing the minimum width for ReLU networks to approximate functions on compact domains
method Analyzing the minimum width for Lp approximation of Lp functions from [0,1]d to Rdy using ReLU-like activation functions result The minimum width for Lp approximation on a compact domain is exactly max{d_x, d_y, 2} for ReLU-like activation functions Abstract: Shows nonexistence of Courant-type nodal domain bounds for eigenfunctions of Dirichlet-to-Neumann operator.
problem Courant-type nodal domain bounds for eigenfunctions of Dirichlet-to-Neumann operator.
method Constructs a metric on a compact manifold to demonstrate the nonexistence of Courant-type bounds.
result Provides a negative answer to the existence of Courant-type nodal domain bounds.
Solves complex Monge-Ampère equation for measures with pluripolar parts.
problem Characterizing measures with complex Monge-Ampère equation solutions.
method Solves for measures with a pluripolar part in compact Kähler manifolds.
result Generalizes classical results in bounded hyperconvex domains.
3-manifold groups can only have convex co-compact representations if they are geometric or hyperbolic.
problem Understanding which 3-manifold groups can have convex co-compact representations.
method Analyzing representations of 3-manifold groups into projective general linear group, focusing on convex co-compactness.
result Fundamental groups of closed irreducible orientable 3-manifolds can only admit convex co-compact representations if they are geometric or hyperbolic.
We prove the existence of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in some compact Riemannian manifolds of dimension n≥2, with volume close to the volume of the manifold. If the first (positive) eigenfunction φ0 of the Laplace-Beltrami operator over the manifold is a nonconst…
Given a smooth nonfocal compact Riemannian manifold, we show that the so-called Ma--Trudinger--Wang condition implies the convexity of injectivity domains. This improves a previous result by Loeper and Villani.
The reduction of biharmonic maps equation in terms of the Maurer-Cartan form for all smooth map of any compact Riemannian manifolds into a compact Lie group with bi-invariant Riemannian metric is obtained. By this formula, all the biharmonic curves into a compact Lie group and all biharmonic maps from a 2-dimensional o…
A solution to the heat equation between Riemannian manifolds, where the domain is compact and possibly has boundary, will not leave a compact and locally convex set before the image of the boundary does.
In this paper, we shall give a lower diameter bound for compact domain manifolds of shrinking Ricci-harmonic solitons. Our result may be regarded as a generalization to Ricci-harmonic geometry of the recent works by Fernández-López and García-Río (Q. J. Math. 61, 319--327, 2010), Futaki and Sano (Asian J. Math. 17, 17-…
We prove uniqueness of solutions to complex Monge-Ampère equations for small temperature.
problem Proving uniqueness of solutions to complex Monge-Ampère equations.
method Local and global analysis of bounded hyperconvex domains and compact complex manifolds.
result Uniqueness of solutions confirmed for small temperature parameters.
Sharp lower bound for p-Laplacian eigenvalue on non-compact manifolds.
problem Estimating eigenvalues of p-Laplacian on non-compact manifolds. method Sharp lower bound established through domain properties and curvature conditions.
result Sharp lower bound for the first Dirichlet eigenvalue of p-Laplacian. For convex domains, automorphism group and limit set properties are described.
problem Characterize the automorphism group and limit set of convex domains.
method Detailed analysis of automorphism group structure and limit set properties for convex domains with C1,ε boundary. result The automorphism group has finitely many components and the limit set is homeomorphic to a sphere.
Generalizes equivariance and convolution to compact groups for neural networks.
problem Ensuring equivariance in neural networks for various domain actions.
method Representation theory and noncommutative harmonic analysis.
result Convolution is necessary and sufficient for equivariance to compact group actions.
We investigate different concentration-compactness phenomena related to the Q-curvature in arbitrary even dimension. We first treat the case of an open domain in R2m, then that of a closed manifold and, finally, the particular case of the sphere S2m. In all cases we allow the sign of the Q-curvature to vary, …
New invariant for hyperbolic surfaces, geometric criterion for domains.
problem Geometric criterion for bounded domains in complex plane.
method Renormalized volume type invariant on hyperbolic surfaces.
result New geometric criterion for bounded domains in complex plane.
We consider nonlinear gauged sigma-models with Kahler domain and target. For a special choice of potential these models admit Bogomolny (or self-duality) equations -- the so-called vortex equations. We find the moduli space and energy spectrum of the solutions of these equations when the gauge group is a torus T^n, the…