Study on evolving non-compact graphs by powers of Gauss curvature.
problem Evolution of non-compact graphs by powers of Gauss curvature.
method Evolution study of convex complete non-compact graphs by positive powers of Gauss curvature.
result Initial graph evolves by any positive power of Gauss curvature for all time, even if not differentiable.
The Qk flow on graphs evolves them up to a time T.
problem Evolution of graphs by Qk curvature flow. method Proving evolution of complete non-compact graphs up to time T.
result Complete graphs evolve by Qk curvature up to time T. Study shows long-term existence of IMCF on non-compact graphs in hyperbolic space.
problem Long-term existence of IMCF on non-compact graphs in hyperbolic space.
method Investigation of IMCF on bounded graphs over horospheres, use of cutoff functions, and development of a non-compact ODE maximum principle.
result Long time existence of IMCF on non-compact graphs in hyperbolic space.
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.
Minimal graphs over non-compact domains in 3-manifolds solved with estimates and uniqueness results.
problem Solving minimal graphs over non-compact domains in 3-manifolds with a Killing vector field.
method Killing Submersion, Dirichlet problem, Collin-Krust estimates, uniqueness results, removable singularities.
result General Collin-Krust type estimates and uniqueness results for minimal Killing graphs.
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.
Upper bounds for Laplace eigenvalues on non-compact manifolds.
problem Proving upper bounds for Laplace eigenvalues on non-compact manifolds.
method Using geometric data and properties of Cartan-Hadamard manifolds.
result Upper bounds for Laplace eigenvalues in terms of k2 and specific geometric data. 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.
The paper studies the connectedness of a graph's boundary for surfaces.
problem Understanding the topology of the Gromov boundary of fine curve graphs for surfaces.
method Proved a bounded geodesic image theorem, used to show linear connectivity of the Gromov boundary.
result The Gromov boundary of fine curve graphs for surfaces is linearly connected.
In this paper we extend to non-compact Riemannian manifolds with boundary the use of two important tools in the geometric analysis of compact spaces, namely, the weak maximum principle for subharmonic functions and the integration by parts. The first one is a new form of the classical Ahlfors maximum principle whereas …
The paper studies homeotopy groups of leaf spaces for specific foliations.
problem Identifying homeotopy groups of leaf spaces for non-compact surfaces with non-compact leaves.
method Identifying homeotopy groups with automorphisms of graphs and showing induced homomorphisms.
result The induced homomorphism between homeotopy groups is either injective or has a kernel of Z_2.
Estimates for graph embeddings into symmetric spaces derived from coarse geometry.
problem Estimating optimal volume of graph embeddings into symmetric spaces.
method Coarse geometric thick embeddings and wiring techniques.
result Optimal and lower bounds for graph embeddings in symmetric spaces of different ranks.
Study examines mean curvature flow on graphs of maps between manifolds.
problem Investigating mean curvature flow on graphs of maps between manifolds with bounded geometry.
method Investigates the mean curvature flow of graphs of smooth length-decreasing maps f:RmoN. result Uniform decay estimates for all derivatives of order ≥2 of ft along the flow. Graph manifolds can be ends of negatively curved Riemannian manifolds.
problem Characterizing graph manifolds as ends of negatively curved Riemannian manifolds.
method Constructing a complete Riemannian metric on RimesM with negative curvature. result Graph manifolds appear as ends of 4-dimensional Riemannian manifolds with negative curvature.
Ancient curve flows classified into specific types.
problem Classifying ancient finite-entropy curve shortening flows.
method Proving flow types through mathematical analysis.
result Ancient flows are one of several specific types.
Extends canonical measures to metric graphs and proves a generalized Kazhdan's theorem.
problem Understanding limiting measures on metric graphs and their relation to hyperbolic measures.
method Introducing hyperbolic measures on universal covers of metric graphs and proving a generalized Kazhdan's theorem.
result All limiting measures on metric graphs satisfy a Gauss-Bonnet formula, interpreted as a trace formula.
Introduce new boundary mass for asymptotically flat half-manifolds
problem Define boundary mass for asymptotically flat half-manifolds
method Introduce new boundary mass
result Define boundary mass for asymptotically flat half-manifolds
Classifies π1-injective maps between non-compact surfaces.
problem Characterizing maps with injective fundamental groups.
method Proper homotopy classification of maps.
result All π1-injective proper maps are classified. New theorems on non-compact manifolds using dynamical methods.
problem Proving divergence theorems on non-compact Riemannian manifolds.
method Dynamical approach to complete non-compact Riemannian manifolds.
result New divergence theorems on complete non-compact Riemannian manifolds.
Paper proves a theorem for Higgs bundles on certain non-compact manifolds.
problem Proving a generalized Donaldson-Uhlenbeck-Yau theorem for Higgs bundles.
method Generalized Donaldson-Uhlenbeck-Yau theorem on Higgs bundles over non-compact Gauduchon manifolds.
result Proved a generalized Donaldson-Uhlenbeck-Yau theorem for Higgs bundles.
Study Szegő kernel on non-compact CR manifolds with specific conditions.
problem Analyzing Szegő kernel on non-compact CR manifolds.
method Establish Szegő kernel asymptotic expansions on non-compact strictly pseudoconvex CR manifolds with transversal CR R-action under natural geometric conditions. result Szegő kernel asymptotic expansions established on non-compact CR manifolds.
Study on anisotropic curvature flow for noncompact convex hypersurfaces.
problem Anisotropic curvature flow of noncompact convex hypersurfaces.
method Flow of complete noncompact convex hypersurfaces with anisotropy determined by a Wulff shape.
result The flow exists for all positive time for initial conditions.
From a graph G with constant valency v and a (non-compact) manifold C with v boundary components, we build a G-periodic manifold M. This process gives a class of topologically infinite manifolds which generalizes periodic manifolds and includes all riemannian coverings with finitely generated deck-group. Ou…
Condition for Anosov flows on non-compact manifolds without conjugate points.
problem Conditions for Anosov flows on non-compact manifolds.
method Formulated a condition for complete, connected and non-compact Riemannian manifolds.
result No conjugate points in geodesic flow if Anosov.
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. Strong rigidity proven for non-compact surfaces.
problem Proving rigidity of non-compact surfaces.
method Generalized result showing proper homotopy to homeomorphism.
result All non-compact orientable surfaces, except plane and punctured plane, are topologically rigid.
Paper establishes equality for non-compact manifolds.
problem Equality between two types of indices on non-compact manifolds.
method Study of Fredholm property of elliptic boundary value problems on non-compact manifolds.
result Establishes equality between Hochs-Mathai and Atiyah-Patodi-Singer indices for non-compact manifolds.
In this paper we prove several quantitative rigidity results for conformal immersions of surfaces in Rn with bounded total curvature. We show that (branched) conformal immersions which are close in energy to either a round sphere, a conformal Clifford torus, an inverted catenoid, an inverted Enneper's minim…
Non-compact flow lines for scalar curvature prescription on manifolds.
problem Non-compactness in scalar curvature prescription on manifolds.
method Gradient flow analysis of scalar curvature on Riemannian manifolds.
result A modification of the gradient flow leads to compact flow lines.
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.
The paper finds infinitely many magnetic geodesics on non-compact manifolds.
problem Existence and multiplicity of periodic orbits of magnetic flows.
method Morse theory applied to non-compact manifolds with energy levels above the Mañé critical value.
result Infinitely many noncontractible closed magnetic geodesics found.
Paper solves Minkowski problem for non-compact convex sets with asymptotic boundary conditions.
problem Solving Minkowski problem for non-compact convex sets with asymptotic boundary conditions.
method Combining covolume, Hadamard variational formula, and geometric interpretation.
result Solved Minkowski problem for non-compact convex sets under asymptotic conditions.
Extends soliton theory to non-compact cases.
problem Generalized solitons in non-compact settings.
method Place conditions on vector field and curvature, use tensor properties.
result Non-compact q-solitons are stationary and q-flat. Paper calculates homotopy types of non-compact surfaces using groupoids.
problem Computing homotopy types of non-compact foliated surfaces.
method Application of van Kampen theorem for groupoids.
result Computation of homotopy types for specific non-compact surfaces.
The paper establishes a duality between non-compact and compact symmetric pairs.
problem Understanding the relationship between non-compact and compact symmetric pairs.
method Developed a duality theorem between non-compact pseudo-Riemannian semisimple symmetric pairs and commutative compact semisimple symmetric triads.
result Explicit description of a one-to-one correspondence between non-compact and compact symmetric pairs.
Paper proves non-compact inaudibility of symmetry and commutativity.
problem Proving inaudibility of symmetry and commutativity in non-compact settings.
method Using isospectral pairs of generalized Heisenberg groups.
result Proved inaudibility of weak symmetry and commutativity.
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 finds solutions for complex problems on non-compact manifolds.
problem Solving fully nonlinear Yamabe-type problems on non-compact manifolds.
method Existence results for a class of problems, considering both positive and negative cases.
result Explicit examples of manifolds satisfying the hypotheses of the theorems.
Proves functoriality in geometric quantization for non-compact spin-c manifolds.
problem Non-compact spin-c manifolds in geometric quantization.
method Formal geometric quantization procedure for spin-c manifolds.
result Functoriality proven in spin-c setting.
Holonomy map is a local biholomorphism for parabolic projective structures on non-compact surfaces.
problem Holonomy map properties for parabolic projective structures on non-compact surfaces.
method Proving the holonomy map is a local biholomorphism.
result Holonomy map is a local biholomorphism for parabolic projective structures on non-compact surfaces.
Study non-compact surfaces formed by gluing strips and their foliations.
problem Understanding foliations on non-compact surfaces formed by gluing strips.
method Examined surfaces formed by gluing strips and studied the resulting foliations.
result The identity path component of the group of homeomorphisms of the foliation is contractible.
Positive mass theorem for non-smooth metrics on flat manifolds with corners.
problem Proving a positive mass theorem for non-smooth metrics on asymptotically flat manifolds with non-compact boundary.
method Proves a positive mass theorem for metrics that are only continuous across a compact hypersurface.
result Obtains a positive mass theorem on manifolds with non-compact corners.
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.
New theorem on geodesics on non-compact manifolds.
problem Existence of geodesics with specific properties.
method Analyzing geodesics with local homology in maximal degree.
result Infinitely many closed geodesics on non-compact manifolds.
Compact curve solution emerges from non-compact curve.
problem Constructing solutions from non-compact curves.
method Slingshot solution to curve shortening flow.
result Compact embedded solution exists for a finite time.
We study integral geometric properties of non-compact harmonic spaces.
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.
Defines geometric quantization for non-compact Hamiltonian torus manifolds using index theory.
problem Geometric quantization for non-compact Hamiltonian torus manifolds.
method Deformation of Dirac operator along group orbits, localization to lattice points.
result Geometric quantization is independent of the choice of polarization.