Study Cauchy data spaces for non-compact manifolds to understand index theory.
problem Understanding the Atiyah-Patodi-Singer index on non-compact manifolds.
method Using maximal domain on manifolds with non-compact boundary for strongly Callias-type operators.
result New insights into Cauchy data spaces and Atiyah-Patodi-Singer index.
Study extends ODE Maximum Principle to non-compact hypersurfaces in hyperbolic space.
problem Analyzing long-term behavior of IMCF on non-compact hypersurfaces.
method Extends ODE Maximum Principle to non-compact hypersurfaces using Omari-Yau maximum principle at infinity.
result Showed long-time existence and asymptotic convergence of IMCF to horospheres.
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.
Hodge decomposition extended to H1 space on non-compact manifolds.
problem Extending Hodge decomposition to non-compact manifolds and Sobolev spaces.
method Generalization of Hodge decomposition to H1 space on non-compact manifolds of nonpositive curvature. result Hodge decomposition established for H1 space on non-compact manifolds of nonpositive curvature. The paper studies spaces of non-compact real algebraic curves and their uniformisation.
problem Understanding the spaces of non-compact real algebraic curves and their uniformisation.
method Construction of spaces of non-compact real algebraic curves and description of their connected components using Fuchsian groups.
result Any connected component of the spaces of non-compact real algebraic curves is homeomorphic to a quotient of a finite-dimensional real vector space by a discrete group.
New examples of isoparametric families on non-compact symmetric spaces.
problem Constructing isoparametric families with non-austere focal sets.
method New extension method from Euclidean spaces to symmetric spaces of non-compact type.
result First examples of isoparametric families on non-compact symmetric spaces.
We study integral geometric properties of non-compact harmonic spaces.
The paper studies HKKN stratifications for non-compact spaces and proves convexity properties.
problem Proving convexity properties of moment maps for non-compact subsets.
method Algebraic and analytical study of HKKN stratifications for a vector space and compact Kähler manifold, then applying to non-compact subsets.
result Convexity properties of moment maps for invariant subsets are proven.
Volume comparison theorem for rank 1 symmetric spaces proved.
problem Volume comparison for symmetric spaces of non-compact type.
method Normalized Ricci--DeTurck flow to analyze volume functional and derive monotonicity properties.
result Volume comparison theorem established for rank 1 symmetric spaces of non-compact type.
Logarithmic Sobolev inequality proven for non-compact self-shrinkers.
problem Establishing a logarithmic Sobolev inequality for non-compact self-shrinkers.
method Using Alexandrov-Bakelman-Pucci (ABP) method to prove the inequality for Euclidean space, then applying this method to non-compact self-shrinkers.
result Optimal logarithmic Sobolev inequality for complete, non-compact, properly embedded self-shrinkers.
The paper studies how certain surfaces evolve in space-time.
problem Preserving the space-like condition of non-compact hypersurfaces.
method Prescribed mean curvature flow in generalized Robertson-Walker spaces.
result The flow preserves space-like condition and exists for infinite time.
Bonahon's method for compactifying Teichmüller space extended to non-compact surfaces.
problem Compactifying Teichmüller space for non-compact finite area surfaces.
method Using geodesic currents, extending Bonahon's construction.
result The method applies to surfaces of finite area.
Polynomial growth bounds for eigenfunctions on non-compact spaces.
problem Growth of eigenfunctions on non-compact spaces with Gaussian weight.
method Analyzing non-compact manifolds with Gaussian weight and drift Laplacian.
result Showed same polynomial growth bounds as Euclidean space for general tensors.
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. In this paper, we prove that full irreducible curvature-adapted isoparametric submanifolds of codimension greater than one in a symmetric space of non-compact type are principal orbits of Hermann actions on the symmetric spaces under certain conditions.
Researchers derived heat kernel expansions for non-compact spaces using Witten deformation.
problem Heat kernel expansions on non-compact spaces, especially for Witten Laplacians.
method Introduced parabolic distance and used it to derive asymptotic expansions.
result Derived an asymptotic expansion of trace of heat kernel for small-time t. The paper develops new methods to study sharp isoperimetric properties on complex spaces.
problem Sharp isoperimetric comparison on non-collapsed spaces with lower Ricci bounds.
method Original argument to estimate first and second variation of the area for isoperimetric sets, avoiding regularity theory.
result Generalizes results for smooth and non-compact manifolds, Alexandrov spaces, and convex bodies.
We investigate Inverse Mean Curvature Flow (IMCF) of non-compact hypersurfaces in hyperbolic space. Specifically, we look at bounded graphs over horospheres in Hn+1 and show long time existence of the flow. Along the way many important local estimates as well as global estimates are obtained. In addition,…
New spectral analysis on non-compact spaces.
problem Analyzing pseudo-Riemannian locally symmetric spaces.
method Initiating spectral analysis beyond classical settings.
result Recent results in non-compact spaces.
We construct new explicit toric scalar-flat K{ä}hler ALE metrics on weighted projective spaces of non-compact type, which we use to obtain smooth extremal K{ä}hler metrics on appropriate resolutions of orbifolds. In particular, we obtain new extremal metrics certain resolutions of weighted projective spaces of compact …
The paper describes a geometric Schottky group for a non-compact hyperbolic surface with infinite genus.
problem Describing a geometric Schottky group for a non-compact hyperbolic surface with infinite genus.
method Constructing a hyperbolic polygon with an infinite number of sides and identifying sides in pairs with Mobius transformations.
result A precise description of the infinite set of generators of a Fuchsian (geometric Schottky) group.
Constructs real algebraic functions with both compact and non-compact preimages.
problem Finding real algebraic functions with specific preimage properties.
method Explicit construction of real algebraic functions.
result Demonstrates real algebraic functions on non-compact manifolds with non-compact preimages.
We show that compact, locally symmetric spaces of non-compact type have positive simplicial volume. This gives a positive answer to a question that was first raised by Gromov in 1982. We provide a summary of results that are known to follow from positivity of the simplicial volume.
The paper classifies submanifolds in symmetric spaces without analyticity.
problem Classifying submanifolds in symmetric spaces of non-compact type.
method Building theory and analysis of reflective focal submanifolds.
result Submanifolds are principal orbits of Hermann type actions.
The purpose of this article is to prove existence of mass minimizing integral currents with prescribed possibly non-compact boundary in all dual Banach spaces and furthermore in certain spaces without linear structure, such as injective metric spaces and Hadamard spaces. We furthermore prove a weak∗-compactness theo…
No spin structures found in a hyperbolic 4D space.
problem Finding hyperbolic 4-manifolds without spin structures.
method Constructed a non-compact, orientable, hyperbolic 4-manifold.
result Demonstrated existence of a hyperbolic 4D space without spin structures.
Analytic torsion defined for non-compact Lie groups and discrete subgroups.
problem Defining and calculating analytic torsion for non-compact Lie groups and their discrete subgroups.
method Localised analytic torsion and relative analytic torsion defined for Lie groups of type I, using representations and discrete subgroups.
result Relative analytic torsion of (G,Γ) coincides with Lott L2 analytic torsion of a covering space. Develops sublinear Morse theory in symmetric spaces.
problem Understanding sublinear Morse properties in symmetric spaces.
method Theory of sublinearly Morse boundary and lemma in higher rank symmetric spaces.
result Proves sublinear Morse lemma in higher rank symmetric spaces.
Study eta invariant on non-compact manifolds with positive scalar curvature.
problem Proving geometric formulas and index theorems for uniformly positive scalar curvature metrics.
method Using Dirac-Schrödinger operators and relative eta invariant.
result New geometric formula for spectral flow and index formula for uniformly positive scalar curvature metrics.
Study differential operators on non-compact harmonic manifolds, finding conditions for radial fundamental solutions and dense heat-semigroups.
problem Conditions for differential operators on non-compact harmonic manifolds to have specific properties.
method Analyzing the algebra of differential operators, their commutation properties, and using geometric averages.
result Algebra of differential operators on non-compact harmonic manifolds has specific properties related to radial fundamental solutions and dense heat-semigroups.
The paper generalizes spectral section concepts to non-compact spaces.
problem Generalizing spectral sections to non-compact base spaces.
method Generalization to arbitrary base spaces, applications to cobordism theorems, investigation of Riesz continuity.
result If a family of operators has a spectral section, it is Riesz continuous.
We formulate a conjecture that arithmetic locally symmetric manifolds have simple homotopy type, and prove it for the non-compact case. More precisely, we show that, for any symmetric space S of non-compact type without Euclidean de Rham factors, there are constants a=a(S) and d=d(S) such that any non-compact arithmeti…
Study moduli spaces of equivariant minimal surfaces using Higgs bundles.
problem Define and study moduli spaces of equivariant minimal immersions into non-compact symmetric spaces.
method Introduce equivariant minimal immersions, use Higgs bundles to parametrize moduli spaces, and relate to classical invariants.
result Provide details of moduli spaces for RH3 and RH4 using Higgs bundle parametrization. Constructs explicit p-harmonic functions on specific Lie groups.
problem Finding explicit p-harmonic functions on a specific class of Lie groups.
method Constructs explicit p-harmonic functions on rank-one Lie groups of Iwasawa type.
result Proves existence of proper p-harmonic functions on these groups.
In this paper, we obtain a Cartan type identity for curvature-adapted isoparametric hypersurfaces in symmetric spaces of compact type or non-compact type. This identity is a generalization of Cartan-D'Atri's identity for curvature-adapted(=amenable) isoparametric hypersurfaces in rank one symmetric spaces. Furthermore,…
The paper extends rigidity results to non-compact domains and infinite energy maps.
problem Rigidity of maps in non-compact domains with specific conditions.
method Analyzing homomorphisms and pluriharmonic maps between different geometric spaces.
result Existence of ρ-equivariant pluriharmonic maps of infinite energy under certain conditions. In this paper, we develop a loop group description of harmonic maps F:M→G/K ``of finite uniton type", from a Riemann surface M into inner symmetric spaces of compact or non-compact type. This develops work of Uhlenbeck, Segal, and Burstall-Guest to non-compact inner symmetric spaces. To be mo…
In this paper, we introduce a deformation analysis of index theory over non compact manifolds, by use of new functional spaces which are the reduced version of Sobolev spaces. It allows to construct Fredholm theory for elliptic differential operators over non compact spaces which possibly do not have closed range with …
Study shows Sobolev functions on non-compact manifolds can't be approximated by smooth compactly supported functions.
problem Sobolev functions on non-compact manifolds cannot be approximated by smooth compactly supported functions.
method Analysis of Sobolev spaces on non-compact manifolds.
result Proves the failure of the density of smooth compactly supported functions in Sobolev spaces on non-compact manifolds.
For 1≤q≤n−2, we provide explicit examples to demonstrate non-compactness of the Neumann operator for the Kohn Laplacian acting on L2 (0,q)-forms on the unit ball in (2n+1)-dimensional Heisenberg space.
Assigns compact set distance-like functions to non-compact geodesic spaces.
problem Assigning distance-like functions to compact sets in non-compact geodesic spaces.
method Assigns each compact set a distance-like function and studies the pseudo-metric on the space of compact subsets.
result Obtains a pseudo-metric on the space of compact subsets that is less than the Hausdorff distance.
New example shows non-compact manifolds can lack Lp-Calderón-Zygmund inequalities.
problem Exploring Lp-Calderón-Zygmund inequalities on non-compact manifolds. method Developed a concrete example using local deformations of metrics.
result Found a non-compact manifold without Lp-Calderón-Zygmund inequalities. Paper develops a new method for harmonic maps into symmetric spaces.
problem Tackles harmonic maps of finite uniton number into symmetric spaces.
method Uses loop group description and DPW theory to transform results.
result Establishes a 1-1 relation between harmonic maps and normalized potentials.
Develops Gaussian processes on non-compact Lie groups.
problem Invariance to symmetries in non-Euclidean spaces.
method Constructive techniques for stationary Gaussian processes.
result Makes non-Euclidean Gaussian processes compatible with standard software.
Inverts operators near parallel Ricci metric on non-compact manifolds.
problem Inverting curvature operators near a parallel Ricci metric.
method Local invertibility of operators in Sobolev spaces.
result Operators are locally invertible near a parallel Ricci metric.
In this paper we introduce two new methods for constructing harmonic morphisms from solvable Lie groups. The first method yields global solutions from any simply connected nilpotent Lie group and from any Riemannian symmetric space of non-compact type and rank r≥3. The second method provides us with global solutio…
The paper proves geodesic balls maximize the first Steklov eigenvalue in non-compact symmetric spaces.
problem Characterizing geodesic balls in non-compact rank one symmetric spaces.
method Utilizing a weighted isoperimetric inequality on harmonic manifolds, the paper proves the maximization of the first Steklov eigenvalue.
result Geodesic balls uniquely maximize the first Steklov eigenvalue among domains of fixed volume in non-compact rank one symmetric spaces.
It is known that principal orbits of Hermann actions on a symmetric space of non-compact type are curvature-adapted isoparametric submanifolds having no focal point of non-Euclidean type on the ideal boundary of the ambient symmetric space. In this paper, we investigate the mean curvature flows for such a curvature-ada…