Study geometry and topology of manifolds with radial curvature bounds.
problem Understanding manifolds with specific curvature constraints.
method Systematic investigation of manifolds with integral radial curvature bounds.
result Obtained many interesting and important conclusions about manifold geometry and topology.
For non-compact manifolds with boundary we prove that bounded geometry defined by coordinate-free curvature bounds is equivalent to bounded geometry defined using bounds on the metric tensor in geodesic coordinates. We produce a nice atlas with subordinate partition of unity on manifolds with boundary of bounded geomet…
Study of collapsed manifolds with bounded Ricci curvature and non-collapsed universal cover.
problem Understanding collapsed manifolds with specific Ricci curvature properties.
method Ricci flow techniques applied to non-collapsed universal cover.
result Partial extension of nilpotent structural results to global Ricci bounded covering geometry.
3-manifolds with positive scalar curvature and bounded geometry are contractible.
problem Characterizing 3-manifolds with positive scalar curvature and bounded geometry.
method Maximal weak solution to inverse mean curvature flow.
result Complete contractible 3-manifolds with positive scalar curvature and bounded geometry are R3. Quantizes symplectic manifolds with bounded geometry using Berezin-Toeplitz method.
problem Quantization of symplectic manifolds with bounded geometry.
method Berezin-Toeplitz quantization theory.
result Correct semiclassical limit achieved.
Generalizes tools for studying collapsed manifolds to new geometry.
problem Studying collapsed manifolds with bounded sectional curvature.
method Generalizes fibration and stability theorems for compact group actions on manifolds with local bounded Ricci covering geometry.
result Two generalized results used in Xiaochun Rong's work on almost flat manifolds.
We prove a persistence result for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. The bounded geometry of the ambient manifold is a crucial assumption in order to control the uniformity of all estimates throughout the proof.
In this paper, we investigate analytical and geometric properties of certain non-compact boundary-manifolds, namely manifolds of bounded geometry. One result are strong Bochner type vanishing results for the L^2-cohomology of these manifolds: if e.g. a manifold admits a metric of bounded geometry which outside a compac…
The paper establishes lower bounds on injectivity radius and constructs metrics with bounded geometry.
problem Establishing lower bounds on the normal injectivity radius of hypersurfaces and constructing metrics with bounded geometry on manifolds with boundary.
method Pointwise lower estimates and constructions of metrics with bounded geometry.
result The construction of metrics with bounded geometry on arbitrary manifolds with boundary.
Optimization rates improved for manifolds with bounded geometry.
problem Optimizing functions on manifolds with bounded geometry.
method Riemannian gradient descent and dynamic trivialization algorithm.
result Curvature-dependent convergence rates computed explicitly for common manifolds.
The study connects polyhedral manifolds to Riemannian ones with geometric bounds.
problem Connecting polyhedral manifolds to Riemannian manifolds with geometric constraints.
method Using a theorem by C. Lange and B. Bowditch, the study bounds the curvature and injectivity radius of Riemannian manifolds.
result Polyhedral manifolds with bounded geometry are bi-Lipschitz homeomorphic to Riemannian manifolds with controlled curvature and injectivity radius.
The study limits the number of specific foliations with bounded geometry.
problem Bounding the number of isoparametric foliations with bounded geometry.
method Proving finitely many foliations with specific properties and constructing infinite families of non-diffeomorphic foliations.
result There are only finitely many isoparametrically foliated closed connected Riemannian manifolds with bounded geometry, up to foliated diffeomorphism.
The paper extends Yamabe flow results to non-compact manifolds with bounded geometry.
problem Yamabe flow convergence issues on manifolds with infinite volume.
method Curvature-normalized Yamabe flow for manifolds with bounded geometry.
result Long-time existence and convergence of the flow for negative scalar curvature.
We prove a persistence result for noncompact normally hyperbolic invariant manifolds in the setting of Riemannian manifolds of bounded geometry. Bounded geometry of the ambient manifold is a crucial assumption required to control the uniformity of all estimates throughout the proof. The Ck,α-smoothness result is o…
New uniform K-theory and Poincare duality established for manifolds.
problem Developing a new framework for K-theory and K-homology.
method Constructing uniform K-homology, defining external and cap products, proving homotopy invariance and Poincare duality.
result Established Poincare duality between uniform K-theory and uniform K-homology on spin-c manifolds.
The paper proves wellposedness of flows on manifolds with bounded geometry.
problem Analyzing wellposedness of nonlinear flows on manifolds of bounded geometry.
method Establishing conditions for the operator to generate an analytic semigroup, proving existence of resolvent, and using geometric microlocal calculus.
result Wellposedness of nonlinear flows on manifolds of bounded geometry is proven.
We characterize functions which are growth types of Riemannian manifolds of bounded geometry.
Paper proves inequalities for forms on sub-Riemannian manifolds.
problem Establishing inequalities for differential forms on sub-Riemannian contact manifolds.
method Using structure of Rumin's complex, Sobolev-Gaffney inequality for Heisenberg groups, and geometric properties.
result Gaffney type inequality in Sobolev spaces for differential forms on sub-Riemannian contact manifolds with bounded geometry.
A surgery classification theory is introduced for manifolds of bounded geometry up to quasi-isometry. The Borel conjecture for this theory is proven for flat Euclidean space.
The paper proves rigidity results for Einstein manifolds with specific geometric constraints.
problem Understanding the rigidity of Einstein manifolds under bounded covering geometry.
method Analyzing Einstein manifolds with bounded covering geometry to prove rigidity results.
result Compact Einstein manifolds with specific geometric properties are isometric to space forms.
Study collapsing geometry with Ricci curvature, proving Kähler metrics and Killing structures.
problem Collapsing geometry of Riemannian manifolds with Ricci curvature constraints.
method Locally bounded Ricci covering geometry and Ricci flow smoothing techniques.
result Volume collapsed Calabi-Yau manifolds admit Ricci-flat Kähler metrics and compatible Killing structures.
Proves well-posedness for elliptic problems on domains with singular points.
problem Elliptic boundary value problems on domains with singular points.
method Conformal changes of metric, differential geometry of manifolds with boundary and bounded geometry.
result Proves well-posedness and regularity results for elliptic boundary value problems.
Study on the geometry of limit spaces of manifolds with boundary.
problem Understanding the geometry of limit spaces of manifolds with boundary.
method Developed infinitesimal geometry for limit spaces under curvature and diameter bounds.
result Determined the infinitesimal structure and Hausdorff dimensions of boundary singular sets.
Generalizes fully augmented links to doubled 3-manifolds with geometric bounds.
problem Understanding the geometry of fully augmented links in doubled 3-manifolds.
method Constructing fully augmented links on the reflection surface of doubled 3-manifolds and finding bounds on cusp shapes and volumes.
result Bounds on cusp shapes and volumes of hyperbolic links in doubled 3-manifolds.
Uniformly proves index invariance for signature operators on manifolds.
problem Proving index invariance for signature operators under uniform homotopy.
method Uniform homotopy invariance of Roe index for signature operators.
result Uniform homotopy invariance of Roe index for signature operators.
We prove that simply connected open Riemannian manifolds of bounded geometry, linear growth and sublinear filling growth (e.g. finite filling area) are simply connected at infinity.
Upper bounds on geodesics in hyperbolic manifolds.
problem Bounding the number of shortest closed geodesics in hyperbolic manifolds.
method Using the Selberg trace formula, we derive upper bounds on the number of shortest and primitive geodesics.
result Generalized Parlier's theorem to higher dimensions and uniform bounds for manifolds with bounded geometry.
We study the Yamabe problem on open manifolds of bounded geometry and show that under suitable assumptions there exist Yamabe metrics, i.e. conformal metrics of constant scalar curvature. For that, we use weighted Sobolev embeddings.
New result on energy-genus bounds for 6D symplectic Calabi-Yau manifolds.
problem Energy-genus bounds for pseudo-holomorphic curves in almost complex manifolds.
method Compactness and regularity theorems for J-holomorphic currents.
result Established a new result in dimension 6 for symplectic Calabi-Yau 6-manifolds.
In this paper we develop the geometry of bounded Fréchet manifolds. We prove that a bounded Fréchet tangent bundle admits a vector bundle structure. But the second order tangent bundle T2M of a bounded Fréchet manifold M, becomes a vector bundle over M if and only if M is endowed with a linear connection. As a…
Local Hardy spaces defined for Riemannian manifolds with bounded geometry.
problem Defining Hardy spaces for Riemannian manifolds with specific curvature conditions.
method Using local Riesz transforms and atomic Goldberg-type spaces.
result Atomic Hardy spaces and local Hardy spaces are equivalent on Riemannian manifolds with bounded geometry.
Compactness theorem for manifolds with boundary proved.
problem Proving convergence of manifolds with boundary.
method Reduction to Hamilton's compactness theorem for manifolds without boundary.
result Cheeger-Gromov convergence for a subsequence of manifolds with bounded geometry.
Ricci flow stability on manifolds with bounded geometry ensures convergence to hyperbolic metrics.
problem Stability and convergence of Ricci flow on manifolds with bounded geometry.
method Continuous dependence on initial conditions, sectoriality of Ricci-DeTurck flow generator, and Hölder norm analysis.
result Ricci flow converges to hyperbolic metrics under certain conditions.
The paper defines function spaces on manifolds with bounded or singular geometries.
problem Defining function spaces on manifolds with various geometries.
method Introduces and analyzes Sobolev, Besov, and Bessel potential spaces on uniformly regular and singular Riemannian manifolds.
result Demonstrates maximal regularity for a linear parabolic problem on singular manifolds.
Develops Lefschetz theory for noncompact manifolds.
problem Lefschetz fixed-point theory for noncompact manifolds.
method Introduces uniform bounded cohomology and develops obstruction theory.
result Uniform Lefschetz class vanishes if and only if map is homotopic to a strongly fixed-point free map.
The main theorem states that any complete connected Riemannian manifold of bounded geometry can be isometrically realized as a leaf with trivial holonomy in a compact Riemannian foliated space.
If Pi: M -> B is an onto smooth maximal rank map between complete Riemannian manifolds M and B with bounded geometry, we prove sufficient conditions for M to be roughly isometric to the Riemannian product FxB, where F is a fiber of M.
The study proves the existence and properties of isoperimetric clusters in Riemannian manifolds with bounded geometry.
problem Proving the existence and properties of isoperimetric clusters in Riemannian manifolds with bounded geometry.
method Proved the existence of isoperimetric clusters and compactness theorem for sequence of clusters, introduced Holder continuity of multi-isoperimetric profile.
result Existence and properties of isoperimetric clusters in Riemannian manifolds with bounded geometry.
We prove that the moduli space of complete Riemannian metrics of bounded geometry and uniformly positive scalar curvature on an orientable 3-manifold is path-connected. This generalizes the main result of the fourth author [Mar12] in the compact case. The proof uses Ricci flow with surgery as well as arguments involvin…
The paper extends a theorem to manifolds with local Ricci bounded covering geometry.
problem Understanding collapsed manifolds with specific Ricci curvature properties.
method Extending a theorem from nilpotent fiber bundles to manifolds with local (ρ,v)-bound Ricci covering geometry. result Manifolds with local (ρ,v)-bound Ricci covering geometry are diffeomorphic to infra-nilmanifolds. Liouville theorems extended to graphs with bounded geometry.
problem Ancient solutions of subexponential growth on graphs.
method Extended Mosconi's results to graphs with bounded geometry.
result Nonnegative ancient solutions are stationary and harmonic.
The study extends calibrated geometry to smooth maps and finds energy bounds.
problem Finding energy bounds for smooth maps between Riemannian manifolds.
method Generalizing calibrated submanifolds to smooth maps and applying to energy functional.
result Lower bounds to the energy of smooth maps in homotopy classes.
Let (M,g) be a complete noncompact riemannian manifold with bounded geometry and parallel Ricci curvature. We show that some operators, "affine" relatively to the Ricci curvature, are locally invertible, in some classical Sobolev spaces, near the metric g.
Let M be a Riemannian manifold with a smooth boundary. The main question we address in this article is: "When is the Laplace-Beltrami operator Δ:Hk+1(M)∩H01(M)→Hk−1(M), k∈N0, invertible?" We consider also the case of mixed boundary conditions. The study of this main question lea…
Study compares manifolds with boundary under weighted Ricci curvature bounds.
problem Understand geometric properties of manifolds with boundary under lower weighted Ricci curvature bounds.
method Use lower N-weighted Ricci curvature bounds with ε-range to study comparison geometry. result Conclude splitting theorems and comparison geometric results for inscribed radius, volume, and eigenvalues.
We show that a Kleinian surface group, or hyperbolic 3-manifold with a cusp-preserving homotopy-equivalence to a surface, has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend only on its end invariants. Bounded geometry is a positive lower bound on the leng…
The study extends convergence theorems for Ricci-limit spaces with bounded curvature.
problem Understanding convergence properties of Ricci-limit spaces with bounded curvature.
method Establishing C1,α-regularities and applying Fukaya's fibration theorem. result Optimal generalization of Fukaya's fibration theorem to C1,α limit spaces. The Serre-Swan theorem in differential geometry establishes an equivalence between the category of smooth vector bundles over a smooth compact manifold and the category of finitely generated projective modules over the unital ring of smooth functions. This theorem is here generalized to manifolds of bounded geometry. I…