Survey on gluing constructions under lower curvature bounds.
problem Understanding lower curvature bounds in various geometric contexts.
method Analyzes gluing constructions in smooth and non-smooth settings.
result Provides conjectures and theorems on synthetic lower Ricci curvature bounds.
Unified study of Riemannian and sub-Riemannian geometries with synthetic Ricci curvature bounds.
problem Unified framework for Riemannian and sub-Riemannian geometries.
method Study of gauge metric measure spaces.
result Unified synthetic Ricci curvature lower bounds for both Riemannian and sub-Riemannian structures.
Study proves upper bounds for solutions on Riemannian manifolds.
problem Proving upper bounds for solutions of Leibenson's equation on Riemannian manifolds.
method Proved upper bounds equivalent to a euclidean-type Sobolev inequality.
result Upper bounds for solutions of Leibenson's equation on Riemannian manifolds are equivalent to euclidean-type Sobolev inequalities.
Lower bound found for eigenvalue of hypersurface in Riemannian manifold.
problem Finding bounds for eigenvalues of hypersurfaces in Riemannian manifolds.
method Used minimally embedded hypersurface and Ricci curvature constraints.
result Provided a lower bound for the first eigenvalue.
The paper sets new bounds for Ricci-curvature in submersions and maps.
problem Establishing bounds for Ricci-curvature in submersions and maps.
method Developed upper and lower bounds for Ricci-curvature in submersions and maps, providing geometric characterizations.
result New bounds for Ricci-curvature in submersions and maps have been derived.
Upper bound on geodesic ball volume in Riemannian manifolds.
problem Bounding geodesic ball volume in Riemannian manifolds.
method Techniques to provide an upper bound on geodesic ball volume.
result Upper bound on geodesic ball volume in Euclidean space.
We prove that Berwald spaces whose flag curvature is nowhere vanishing are in fact Riemannian spaces. This means that any Berwald space with flag curvature bounded below by a positive number must be also Riemannian. This rigidity result shows the importance of non-Riemannian examples when imposing flag curvature bounds…
Lower bounds on curvature integral for manifolds with curvature constraints.
problem Bounding curvature integrals under curvature constraints.
method Proving a lower bound for the curvature integral using dimension, upper curvature bounds, and injectivity radius.
result Uniformly bounded below integral of scalar curvature.
Improved bounds on geodesic lengths in Riemannian surfaces.
problem Finding precise lengths of geodesics in Riemannian surfaces.
method Proved curvature-free linear length bounds on geodesics.
result Length of kextth-shortest geodesic is at most 8kd. Uniform curvature bounds for regularized metrics with bounds on Ricci tensor and injectivity radius.
problem Bounding curvature of regularized metrics with constraints on Ricci tensor and injectivity radius.
method Mollification of riemannian metrics, uniform W2,p-harmonic radius bounds, Ricci tensor bounds, injectivity radius bounds. result Uniform estimate on the change of sectional curvature for regularized metrics.
Study finds bounds for fundamental tone on special Riemannian manifolds.
problem Finding bounds for fundamental tones on specific Riemannian manifolds.
method Developed a general lower bound for the fundamental tone of the p-Laplacian.
result Results applied to negatively curved manifolds, warped products, and Riemannian submersions.
Sharp bounds on heat kernel derivatives on incomplete manifolds.
problem Extending bounds on heat kernel derivatives to incomplete Riemannian manifolds.
method Analyzing heat kernels on incomplete Riemannian manifolds with conservative and non-conservative vector fields.
result Sharp bounds on all orders of heat kernel derivatives are established for incomplete manifolds.
New bounds found for nodal sets on special manifolds.
problem Finding bounds for nodal sets on specific types of manifolds.
method Used polynomial upper bounds for eigenfunctions on Gevrey and quasianalytic Riemannian manifolds.
result Established new upper bounds for the size of nodal sets.
Study finds lower bounds for solutions on Riemannian orbifolds.
problem Finding solutions to nonlinear elliptic problems on Riemannian orbifolds.
method Employed the photography method to establish a lower bound.
result Lower bound for the number of solutions in terms of category.
Derives sub-Riemannian Ricci curvature for various manifolds.
problem Calculating Ricci curvature in sub-Riemannian geometry.
method Generalized Gamma z calculus and z--Bochner's formula. result Analytical bounds for sub-Riemannian curvature dimension and log-Sobolev inequalities.
New bounds for low-regularity Riemannian metrics defined via distributional curvature.
problem Establishing curvature bounds for Riemannian metrics of low regularity.
method Introducing a distributional version of sectional curvature for C1 and C0 metrics. result New bounds for low-regularity metrics recover classical bounds in Alexandrov spaces.
This paper examines limits of Riemannian 2-manifolds with bounded curvature.
problem Understanding the limits of Riemannian 2-manifolds with bounded curvature.
method Uniform semi-locally 1-connected sequences of closed connected Riemannian 2-manifolds with bounded total absolute curvature.
result Description of Gromov-Hausdorff limits of the sequences.
We prove that Riemannian metrics with an absolute Ricci curvature bound and a conjugate radius bound can be smoothed to having a sectional curvature bound. Using this we derive a number of results about structures of manifolds with Ricci curvature bounds.
Sharp sub-Gaussian bounds for subsolutions of Trudinger's equation on Riemannian manifolds.
problem Bounding weak subsolutions of Trudinger's equation on Riemannian manifolds.
method Proving sub-Gaussian upper bounds for weak subsolutions.
result The upper bounds are sharp for specific classes of manifolds, including \(\mathbb{R}^{n}\).
Given a metrically complete Riemannian manifold (M,g) with smooth nonempty boundary and assuming that one of its curvatures is subject to a certain bound, we address the problem of whether it is possibile to realize (M,g) as a domain inside a geodesically complete Riemannian manifold (M′,g′) without boundary, by …
The paper converts metric bounds to distance function Hölder bounds and proves compactness theorems.
problem Proving geometric stability results with scalar curvature bounds.
method Transforming Lp bounds to Hölder bounds for distance functions. result Compactness theorems and convergence guarantees for Riemannian manifolds.
The paper proves inequalities for submanifolds in Riemannian manifolds.
problem Proving geometric inequalities for submanifolds in Riemannian manifolds.
method Using Rauch's comparison theorem and first variation formula.
result General Li-Yau inequality applicable in bounded sectional curvature manifolds.
The study examines metrics on Riemannian spaces with bounded properties and finds conditions for Lipschitz and uniform bounds.
problem Investigating bounded rough Riemannian metrics and their properties.
method Analyzing the structure of bounded rough Riemannian metrics and finding conditions for Lipschitz and uniform bounds.
result Weak conditions are identified for Lipschitz and uniform bounds on the metrics.
Sharp bounds on scalar curvature spectrum and rigidity theorems.
problem Understanding scalar curvature bounds and rigidity on manifolds.
method Sharp upper bounds for the bottom spectrum of the Beltrami Laplacian, scalar curvature rigidity theorem.
result Sharp upper bound for the bottom spectrum of the Beltrami Laplacian and scalar curvature rigidity theorem.
We study eigenvalue problems for intrinsic sub-Laplacians on regular sub-Riemannian manifolds. We prove upper bounds for sub-Laplacian eigenvalues λk of conformal sub-Riemannian metrics that are asymptotically sharp as k→+∞. For Sasakian manifolds with a lower Ricci curvature bound, and more generally, for…
Study solves Gel'fand's inverse problem in non-smooth spaces with Ricci curvature bounds.
problem Determining a Riemannian manifold from heat kernel on subsets.
method Analyzes mRCD(K,N) spaces with synthetic Ricci curvature bounds. result Unique solvability of Gel'fand's inverse problem for compact mRCD(K,N) spaces. A leafwise Hodge decomposition was proved by Sanguiao for Riemannian foliations of bounded geometry. Its proof is explained again in terms of our study of bounded geometry for Riemannian foliations. It is used to associate smoothing operators to foliated flows, and describe their Schwartz kernels. All of this is extend…
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.
Willmore-type inequalities for bounded domains in manifolds with curvature bounds.
problem Establishing inequalities for bounded domains in manifolds with curvature bounds.
method Using asymptotic or integral Ricci curvature bounds to establish inequalities.
result Recovering a recent inequality of Jin-Yin.
Estimates for eigenvalues on Riemannian manifolds using classical inequalities.
problem Estimating eigenvalues of the Dirichlet Laplacian on Riemannian manifolds.
method Building on Li-Yau's and Yang's inequalities, deriving upper and lower bounds.
result Explicit estimates on lower bounds for eigenvalues of the Dirichlet Laplacian on projective spaces and their minimal submanifolds.
Characterizes orbifolds with upper curvature bounds as reflectofolds.
problem Understanding orbifolds with upper curvature bounds.
method Characterization through Alexandrov curvature and reflectofolds.
result Quotients of Riemannian manifolds by isometries have locally bounded curvature if and only if they are reflectofolds.
Sharp upper bound found for stable minimal surfaces.
problem Bounding the diameter of stable minimal surfaces.
method Analyzing three-dimensional Riemannian manifolds with specific curvature conditions.
result Sharp upper bound for the diameter of stable minimal surfaces.
Generalized Blaschke rolling theorem for curved spaces.
problem Extending classical theorem to curved spaces.
method Generalization to Riemannian manifolds with bounded curvature.
result Sharp results in arbitrary dimensions, new even in constant curvature spaces.
A classical result by Cheng in 1976, improved later by Besson and Nadirashvili, says that the multiplicities of the eigenvalues of the Schrodinger operator with a smooth potential on a compact Riemannian surface M are bounded in terms of the eigenvalue index and the genus of M. We prove that these multiplicity bounds h…
Under the assumption of the uniform local Sobolev inequality, it is proved that Riemannian metrics with an absolute Ricci curvature bound and a small Riemannian curvature integral bound can be smoothed to having a sectional curvature bound. This partly extends previous a priori estimates of Ye Li (J. Geom. Anal. 17 (20…
The paper bounds radii and curvatures in Riemannian manifolds.
problem Bounding radii and curvatures in Riemannian manifolds.
method Analyzing scalar curvature, injectivity radius, and mean curvature.
result Proves bounds on injectivity and focal radii under specific conditions.
For a fat sub-Riemannian structure, we introduce three canonical Ricci curvatures in the sense of Agrachev-Zelenko-Li. Under appropriate bounds we prove comparison theorems for conjugate lengths, Bonnet-Myers type results and Laplacian comparison theorems for the intrinsic sub-Laplacian. As an application, we consider …
Upper bounds for volume spectrum depend on volume, dimension, and a conformal invariant.
problem Bounding the volume spectrum of Riemannian manifolds.
method Proves upper bounds that depend on volume, dimension, and a conformal invariant.
result Upper bounds for the volume spectrum are established.
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 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.
Lower bounds on average normal curvature for submanifolds in Riemannian domains.
problem Finding bounds on the average normal curvature of submanifolds in Riemannian domains.
method Using an invariant measuring optimal n-trace convexity under unit-gradient normalization. result Lower bounds for the average normal curvature expressed in terms of an invariant.
Study of Lagrangian submanifolds with Riemannian bounds and their metric properties.
problem Understanding the geometry and topology of Lagrangian submanifolds with Riemannian constraints.
method Investigation of metric properties and symplectic structures on spaces of Lagrangian submanifolds with uniform Riemannian bounds.
result There are at most countably many Hamiltonian isotopy classes of exact Lagrangian submanifolds in a Liouville manifold.
The study examines the smoothness of submetries in Riemannian manifolds.
problem Regularity of submetries in Riemannian manifolds and their quotient spaces.
method Analysis of equidistant decompositions and quotient spaces.
result Strata of quotient spaces have curvature bounded from both sides.
The paper studies heat kernels on weighted Riemannian manifolds with curvature bounds.
problem Estimating heat kernels on weighted Riemannian manifolds with lower Ricci curvature bounds.
method Establishing parabolic Harnack inequalities, proving Gaussian bounds for heat kernels, and constructing Li-Yau-type gradient estimates.
result Gaussian upper and lower bounds for the heat kernel, Liouville theorem, uniqueness property, and eigenvalue bounds.
Compactness theorem for Riemannian manifolds with volume and curvature bounds.
problem Investigating the regularity of limit spaces of Riemannian manifolds.
method Local volume growth condition, compactness theorem, different convergence notion.
result Compactness theorem for Riemannian manifolds with Lp curvature bounds and volume growth assumption. The study bounds Riesz transforms on manifolds with controlled curvature.
problem Bounding Riesz transforms on manifolds with controlled curvature.
method Established Lp-boundedness of local covariant Riesz transforms for differential forms. result Calderón-Zygmund estimates for manifolds with bounded Riemannian curvature.
New geometric SDEs and discretizations on Riemannian manifolds with error bounds.
problem Modeling diffusion processes on Riemannian manifolds with geometric SDEs.
method Introduced a new construction of geometric SDEs and provided non-asymptotic error bounds.
result First non-asymptotic error bound for geometric Euler-Murayama discretization.
The paper studies limits of sub-Riemannian Heisenberg manifolds and proves convergence under certain conditions.
problem The study of limits of sub-Riemannian Heisenberg manifolds and their properties.
method Analysis of Gromov-Hausdorff limits with upper and lower bounds on diameter and Popp's measure.
result The non-collapsed limits of sub-Riemannian Heisenberg manifolds converge to a compact Heisenberg manifold.