The paper proves non-existence theorems for certain Riemannian manifolds.
problem Non-existence theorems for specific Riemannian manifolds.
method Proves Liouville-type theorems for complete Riemannian almost product manifolds and special mappings.
result Generalizes similar results for compact manifolds to complete manifolds.
Smooth Riemannian manifolds can be embedded without boundary.
problem Embedding smooth Riemannian manifolds with boundary into complete manifolds without boundary.
method General gluing and conformal-deformation construction.
result Any smooth, metrically complete Riemannian manifold with smooth boundary can be realized as a closed domain into a smooth, geodesically complete Riemannian manifold without boundary.
Discontinuous isoperimetric profile found for a Riemannian manifold.
problem Finding discontinuous isoperimetric profiles in Riemannian manifolds.
method First known example provided.
result Discontinuous isoperimetric profile exists for a Riemannian manifold.
New Einstein metrics found from para-Sasaki-like Riemannian manifolds.
problem Finding new Einstein metrics in Riemannian geometry.
method Cone construction and hyperbolic extension of paracontact paracomplex Riemannian manifolds.
result New complete Einstein para-Sasaki-like Riemannian manifolds with negative scalar curvature.
Uniform boundedness of Riesz transforms on Riemannian manifolds is established with a dichotomy.
problem Establishing uniform boundedness of Riesz transforms on Riemannian manifolds.
method Constructing a complete Riemannian manifold M to demonstrate the dichotomy. result A dichotomy concerning uniform boundedness of Riesz transforms on Riemannian manifolds.
Gradient estimates for solutions to a specific nonlinear equation on Riemannian manifolds.
problem Estimating gradients of solutions to a nonlinear elliptic equation on Riemannian manifolds.
method Analyzing the equation Δu + cu^α = 0 to derive gradient estimates.
result Gradient estimates for positive solutions to the given equation on complete Riemannian manifolds.
Paper proves a Liouville theorem for harmonic functions in complete Riemannian manifolds.
problem Existence of Killing potential in complete Riemannian manifolds.
method Gradient estimation and integral form of Liouville theorem.
result Harmonic functions in complete Riemannian manifolds are constants along geodesics.
Vanishing theorem for harmonic forms on complete foliated manifolds.
problem Existence of nontrivial basic harmonic forms on foliated manifolds.
method Extending vanishing theorems to complete foliated Riemannian manifolds.
result No nontrivial basic harmonic forms on complete foliated Riemannian manifolds.
Though Trudinger-Moser inequalities on compact Riemannian manifolds or Euclidean space are well understood, we know little about them on complete noncompact Riemannian manifolds. In this paper, we established respectively necessary condition and sufficient condition under which Trudinger-Moser inequalities hold on comp…
Proves a Liouville-type theorem for p-Laplacian on manifolds.
problem Proving Liouville-type theorems for p-Laplacian on manifolds.
method Proved a Liouville-type result for the p-Laplacian on complete Riemannian manifolds.
result Proved a Liouville-type theorem for the p-Laplacian on complete non-compact Riemannian manifolds.
Study classifies manifolds with nonpositive curvature and finds conformal Killing forms.
problem Characterizing manifolds with nonpositive curvature operator.
method Classification and vanishing theorems for conformal Killing forms.
result Classification and vanishing results for conformal Killing forms.
The study proves non-existence theorems for Codazzi tensors on Riemannian manifolds.
problem Proving non-existence theorems for Codazzi tensors on Riemannian manifolds.
method Using theorems connecting manifold geometry and subharmonic functions.
result Several Liouville-type non-existence theorems for Codazzi tensors.
Paper studies eigenvalues of a specific operator on Riemannian manifolds.
problem Eigenvalues of a specific operator on Riemannian manifolds.
method Established a general formula for eigenvalues and derived estimates.
result Obtained universal inequalities for the eigenvalues on translating solitons.
Haefliger cohomology characterizes taut foliated manifolds by Haefliger's theorem. We show that Haefliger cohomology characterizes strongly tense foliated manifolds, namely, foliated manifolds which admit a Riemannian metric such that the mean curvature form of the leaves is closed and basic. We show that Haefliger coh…
The study counts cusps on finite volume Riemannian manifolds using volume and decay estimates.
problem Counting cusps on finite volume Riemannian manifolds.
method Volume and decay estimates, nonlinear theory of p-Laplacian, volume comparison theorems.
result An upper bound on the number of cusps based on the volume of the manifold.
Geodesic convexity types differ in Riemannian manifolds.
problem Characterizing geodesic convexity types in Riemannian manifolds.
method Reverse engineering to characterize manifolds with coinciding convexity types.
result Characterized complete manifolds with coinciding geodesic convexity types.
In this short note, we find a new gap phenomena on Riemannian manifolds, which says that for any complete noncompact Riemannian manifold with nonnegative Ricci curvature, if the scalar curvature decays faster than quadratically, then it is Ricci flat.
Discussing Riemannian Hilbert manifolds, their properties, and extending a theorem.
problem Properties and extensions of Riemannian Hilbert manifolds.
method Analyzing singularities, completeness, and homogeneity; extending a theorem.
result Extended a theorem due to Nomizu and Ozeki to infinite dimensional Riemannian Hilbert manifolds.
Paper proves inequality for p-Laplacian eigenvalues on curved spaces.
problem Eigenvalue inequalities for p-Laplacian on curved manifolds.
method Robin boundary conditions, lower Ricci bounds, positive asymptotic volume ratio.
result Bossel-Daners inequality extends to compact submanifolds.
Complete manifolds can be leaves in compact foliations.
problem Understanding the structure of Riemannian manifolds.
method Isometric realization in compact foliated spaces.
result Complete manifolds can be realized as leaves with trivial holonomy.
Study rough Riemannian metrics on manifolds, proving their connectedness and completeness.
problem Understanding the space of all locally elliptic and bounded Riemannian metrics on manifolds.
method Introduced an extended metric space and proved its properties.
result Proved the space of rough Riemannian metrics is complete and connected.
Study harmonic mappings and submanifolds using Bochner technique.
problem Classical theorems in harmonic mappings and submanifolds.
method Generalized Bochner technique.
result New insights into classical theorems.
We show that a complete flat pseudo-Riemannian homogeneous manifold with non-abelian linear holonomy is of dimension at least 14. Due to an example constructed in a previous article by Oliver Baues and the author, this is a sharp bound. Also, we give a structure theory for the fundamental groups of complete flat pseudo…
Uniform Poincaré inequalities on manifolds with bounded Ricci curvature.
problem Establishing uniform Poincaré inequalities on manifolds with specific curvature properties.
method Analyzing manifolds with polynomial growth and bounded Ricci curvature.
result Uniform Poincaré inequalities on manifolds with polynomial growth and bounded Ricci curvature.
New approach for tensor completion using Riemannian manifold optimization.
problem Tensor completion with rank constraint.
method Riemannian manifold preconditioning approach with novel metric.
result Robust algorithms outperform state-of-the-art across various datasets.
Let M be a complete Riemannian manifold possessing a strictly convex Lipschitz continuous exhaustion function. We show that the isoperimetric profile of M is a continuous and non-decreasing function. Particular cases are Hadamard manifolds and complete non-compact manifolds with strictly positive sectional curvatur…
We study Fredholm properties and index formulas for Dirac operators over complete Riemannian manifolds with straight ends. An important class of examples of such manifolds are complete Riemannian manifolds with pinched negative sectional curvature and finite volume.
Extends Hopf-Rinow theorem to semi-Riemannian spacetimes.
problem Generalizing Hopf-Rinow theorem to compact Lorentzian manifolds.
method Develops null distance for proper cone structures and (n−ν,ν)-spacetimes. result Generalizes Hopf-Rinow theorem to a new class of semi-Riemannian manifolds.
The paper tackles the smooth Riemannian extension problem, providing existence results and obstructions.
problem Addressing the problem of extending a Riemannian manifold with smooth boundary to a geodesically complete one.
method Providing three types of results: existence theorems, topological obstructions, and existence results under certain conditions.
result Existence of geodesically complete Riemannian extensions without curvature constraints, various obstructions, and specific existence results under convexity conditions.
In this paper we characterise with the matrix the complete flag of riemannian extension (see définition) on a riemannian compact manifold whose metric is bundlelike for any foliation F_{s} of this flag. This study show us that a foliation of a complete flag of riemannian extension on a riemannian compact manifold whose…
Study proves inequalities for eigenvalues of fourth-order elliptic operators on Riemannian manifolds.
problem Eigenvalue inequalities for fourth-order elliptic operators on Riemannian manifolds.
method Proves inequalities using Payne-Pólya-Weinberger-Yang type for eigenvalues of fourth-order elliptic operators in divergence form on complete Riemannian manifolds.
result Generalizes eigenvalue inequalities for the clamped plate problem to complete Riemannian manifolds.
We generalize for pseudo-Riemannian metrics a classical result of Gallot and Tanno and use it to reprove a recent result of Alekseevsky, Cortes, Galaev and Leistner that decomposable cones over complete closed pseudo-Riemannian manifolds do not exist.
Sharp inequalities for manifolds with nonnegative curvature.
problem Establishing inequalities for manifolds with nonnegative curvature.
method Using the ABP-method, generalizing previous work by Brendle.
result Sharp Sobolev and isoperimetric inequalities for compact domains and submanifolds.
We exhibit 3 families of complete curvature homogeneous pseudo-Riemannian manifolds which are modeled on irreducible symmetric spaces and which are not locally homogeneous. All of the manifolds have nilpotent Jacobi operators; some of the manifolds are, in addition, Jordan Osserman and Jordan Ivanov-Petrova.
Study local curvature estimates and existence of conformal metrics on noncompact manifolds.
problem Deriving local C0-estimates and existence of conformal metrics with prescribed curvature. method Utilizing Aviles-McOwen's result and its nonlinear extension, combined with asymptotic conditions.
result Proved existence of complete conformal metrics with prescribed curvature functions.
Completeness theorem for flat pseudo-Riemannian manifolds of signature (2,2).
problem Completeness of closed flat pseudo-Riemannian manifolds of signature (2,2).
method Geometric reduction and semidirect product constructions.
result Only the entire space R2,2 is divisible by a discrete subgroup of isometries. Classifies ambitoric 4-orbifolds with completeness assumptions.
problem Classifying ambitoric 4-orbifolds with completeness assumptions.
method Developed tools to classify regular ambitoric 4-orbifolds with completeness assumptions.
result Proved a partial classification of compact Riemannian 4-manifolds with a Killing 2-form.
Vanishing theorem for L2-harmonic forms on Riemannian manifolds with parallel 1-form.
problem Proving vanishing of L2-harmonic forms on Riemannian manifolds with a parallel 1-form. method Using L2 Morse-Novikov cohomology and a vanishing theorem. result The L2-harmonic forms on the manifold are identically zero. A simpler proof shows L2-metric completion is CAT(0).
problem Completing Riemannian metrics space.
method Easier proof of existing result by Brian Clarke.
result Completion of Riemannian metrics is CAT(0).
In Finsler geometry the complete lift vector fields have distinguished geometric significance. For example a vector field on a Finsler manifold is said to be conformal if its complete lift is conformal in usual sense. In this work we define a new Riemannian or Pseudo-Riemannian metric on TM derived from a Finsler metri…
This is the author's Ph.D. thesis, submitted to the University of Leipzig. It deals with the L2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold. The main body of the thesis is a description of the completion manifold of metrics with respect to the $L^…
Study characterizes hulls and capacities on Riemannian manifolds, proving isoperimetric inequalities.
problem Characterizing hulls and capacities on Riemannian manifolds.
method Investigates strictly outward minimising hulls and uses p-capacities to recover their areas.
result Sharp isoperimetric inequality on complete noncompact manifolds with nonnegative Ricci curvature.
We obtain upper bounds on the heat content and on the torsional rigidity of a complete Riemannian manifold M, assuming a generalized Hardy inequality for the Dirichlet Laplacian on M.
We give an estimate of the first eigenvalue of the Laplace operator on a complete noncompact stable minimal hypersurface M in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface M has sufficiently smal…
Study Morse functions on manifolds with positive mean curvature.
problem Understanding functions on manifolds with positive mean curvature.
method Construct Morse functions on complete Riemannian manifolds with level hypersurfaces of positive mean curvature.
result If a manifold has no such functions, it contains a complete minimal hypersurface.
The paper proves a Liouville theorem for heat flows on manifolds with specific curvature conditions.
problem Investigating heat flows on manifolds with specific curvature conditions.
method Gradient estimate and Liouville type theorem for ancient solutions.
result Established a Liouville theorem for V-harmonic heat flows. The study proves conditions for complete Riemannian manifolds to be Einstein.
problem Conditions for complete Riemannian manifolds to be Einstein.
method Proving conditions using harmonic curvature and curvature operator of the second kind.
result Complete Riemannian manifolds with specific curvature conditions are Einstein.
In this article, we study complete pseudo-Riemannian manifolds whose cone admits a parallel symmetric 2-tensorfield. The situation splits in three cases: nilpotent, decomposable or complex Riemannian. In the complex Riemannian and decomposable cases we provide a classification. In the nilpotent case, we are able to des…