Study finds new factorable surfaces with non-zero curvature in pseudo-Galilean space.
problem Classifying surfaces with non-zero curvature in pseudo-Galilean space.
method Analyzing factorable surfaces as graphs of product functions.
result New classification results for factorable surfaces with non-zero Gaussian and mean curvature.
No regular algebraic hypersurfaces with non-zero constant mean curvature in Euclidean spaces are found.
problem Existence of regular algebraic hypersurfaces with non-zero constant mean curvature in Euclidean spaces.
method Analyzing polynomials defining hypersurfaces of various degrees and shapes.
result Hyperspheres and round cylinders are the only such hypersurfaces defined by polynomials of degree ≤3.
Estimates the first non-zero eigenvalue using Ricci curvature on graph edges.
problem Estimating the first non-zero eigenvalue of the Laplacian on graph edges.
method Defining edge distance, studying coarse Ricci curvature, and using Jost-Horak's Laplacian definition.
result Obtained an estimate of the first non-zero eigenvalue of the Laplacian by the Ricci curvature for a regular graph.
Statistical Lie algebras with constant curvature are linked to locally conformally Kähler structures.
problem Characterizing Lie algebras with constant curvature and their geometric implications.
method Constructing statistical manifolds and Sasakian structures to relate Lie algebras to locally conformally Kähler structures.
result Statistical Lie algebras of constant curvature correspond to locally conformally Kähler Lie algebras.
Recently, we developed a method for the study of holonomy properties of non-Riemannian Finsler manifolds and obtained that the holonomy group can not be a compact Lie group, if the Finsler manifold of dimension >2 has non-zero constant flag curvature. The purpose of this paper is to move further, exploring the holon…
The holonomy group G of a pseudo-quaternionic-Kählerian manifold of signature (4r,4s) with non-zero scalar curvature is contained in $\Sp(1)\cdot\Sp(r,s)$ and it contains $\Sp(1)$. It is proved that either G is irreducible, or s=r and G preserves an isotropic subspace of dimension 4r, in the last case, ther…
The abstract proves properties of Berwald spaces with non-zero flag curvature.
problem Characterizing Berwald spaces with non-zero flag curvature.
method Analyzes properties of Berwald manifolds with non-zero flag curvature, proving extensions of previous theorems.
result Every Berwald manifold with non-zero flag curvature is Riemannian.
New curvature obstruction for Killing vector fields on Lorentzian manifolds.
problem Existence of timelike or causal Killing vector fields on Lorentzian manifolds.
method New curvature obstruction in terms of timelike or null sectional curvature.
result Extension of Gauss-Bonnet-Chern obstruction to non-zero timelike sectional curvature.
Sphere theorems for p-Laplacian eigenvalues established.
problem Sphere theorems for p-Laplacian eigenvalues.
method Established sphere theorems for p-Laplacian eigenvalues.
result Sphere theorems for p-Laplacian eigenvalues established.
The study analyzes weighted manifolds with curvature bounds, proving eigenvalue estimates and inequalities.
problem Analyzing geometric properties of weighted manifolds under Ricci curvature bounds.
method Develops geometric analysis techniques on weighted Riemannian manifolds with lower 0-weighted Ricci curvature bounds. result Proves eigenvalue estimates for Steklov and ABP inequalities on weighted manifolds.
We complete the reduction of Sasakian manifolds with the non-zero case by showing that Willett's contact reduced space is compatible with the Sasakian structure. We then prove the compatibility of the non-zero Sasakian (in particular, contact) reduction with the reduction of the Kähler (in particular, symplectic) cone.…
Sharp bounds found for the first non-zero Steklov eigenvalue.
problem Finding bounds for the first non-zero Steklov eigenvalue.
method Analyzing star-shaped domains and balls with smooth boundaries.
result Sharp lower and two-sided bounds for the first non-zero Steklov eigenvalue.
Paper finds bounds for Steklov eigenvalues on manifolds.
problem Eigenvalue bounds for Steklov eigenvalues on manifolds.
method Eigenvalue comparison theorems and bounds for the first non-zero eigenvalue of the Wentzell eigenvalue problem.
result Established bounds for Steklov eigenvalues and Wentzell eigenvalues.
For a Riemannian closed spin manifold and under some topological assumption (non-zero A^-genus or enlargeability in the sense of Gromov-Lawson), we give an optimal upper bound for the infimum of the scalar curvature in terms of the first eigenvalue of the Laplacian. The main difficulty lies in the study of the o…
No algebraic 3rd degree hypersurfaces in Euclidean spaces have constant mean curvature.
problem Existence of algebraic hypersurfaces with constant mean curvature.
method Analytical proof.
result No such hypersurfaces exist.
Metric spaces with non-positive curvature have a specific curve isoperimetric property.
problem Understanding spaces with non-positive curvature.
method Proving isoperimetric inequality for metric spaces with Alexandrov curvature.
result Metric spaces with non-positive curvature have a specific curve isoperimetric property.
We construct new explicit compact supersymmetric valid solutions with non-zero field strength, non-flat instanton and constant dilaton to the heterotic equations of motion in dimension six. We present balanced Hermitian structures on compact nilmanifolds in dimension six satisfying the heterotic supersymmetry equations…
We study the 8 natural GL equivariant geometric realization questions for the space of generalized algebraic curvature tensors. All but one of them is solvable; a non-zero projectively flat Ricci antisymmetric generalized algebraic curvature is not geometrically realizable by a projectively flat Ricci antisymmetric tor…
We give an estimate on the lower bound of the first non-zero eigenvalue of the Laplacian for a closed Riemannian manifold with positive Ricci curvature in terms of the in-diameter and the lower bound of the Ricci curvature.
Study on surfaces with constant anisotropic mean curvature in 3D space.
problem Characterizing surfaces with constant anisotropic mean curvature.
method Analyzing uniformly elliptic anisotropic functionals and proving properties of surfaces.
result Characterization of surfaces with constant anisotropic mean curvature.
The paper proves the existence of embedded hypersurfaces of constant mean curvature in manifolds with positive Ricci curvature.
problem Proving the existence of embedded hypersurfaces of constant mean curvature in manifolds with positive Ricci curvature.
method Using the Allen--Cahn min-max scheme with a non-zero constant prescribing function.
result The existence of embedded, closed λ-CMC hypersurfaces with Morse index 1 for any prescribed non-zero constant λ.
Berwald spaces with non-zero flag curvature are Riemannian.
problem Understanding the conditions under which Berwald spaces become Riemannian.
method Analyzing the flag curvature of Berwald spaces and proving their rigidity under certain conditions.
result Berwald spaces with non-zero flag curvature are Riemannian.
The study examines complete conformally flat submanifolds with nullity in Euclidean space.
problem Investigating properties of conformally flat submanifolds with nullity.
method Analyzing the index of relative nullity and scalar curvature to deduce manifold properties.
result Conditions for the manifold to be flat and the immersion to be a cylinder over a submanifold.
Lower bound found for Steklov eigenvalue on curved manifolds.
problem Finding bounds for Steklov eigenvalues on curved spaces.
method Established a new lower bound using geometric curvature conditions.
result Found a new lower bound for the first non-zero Steklov eigenvalue.
Along the line of the Yang Conjecture, we give a new estimate on the lower bound of the first non-zero eigenvalue of a closed Riemannian manifold with negative lower bound of Ricci curvature in terms of the in-diameter and the lower bound of Ricci curvature.
Extends Choi-Wang inequality to Li-Xia affine connections.
problem Eigenvalue bounds for minimal hypersurfaces.
method Positive Ricci curvature and Li-Xia affine connection.
result Established new lower bounds for eigenvalues.
In [29], Plebanski reformulated the anti-self-dual Einstein equations with non-zero scalar curvature as a first order PDE for a connection in an SO(3)-bundle over the four-manifold. The aim of this article is to place this differential equation in a new framework, in which it is both elliptic and a stationary point of …
In Finsler geometry, there are infinitely many models of constant curvature. The Funk metrics, the Hilbert-Klein metrics and the Bryant metrics are projectively flat with non-zero constant curvature. A recent example constructed by the author is projectively flat with zero curvature. In this paper, we introduce a techn…
We show that complete uniform visibility manifolds of finite volume with sectional curvature −1≤K≤0 have positive simplicial volumes. This implies that their minimal volumes are non-zero.
We prove that rationally essential manifolds with suitably large fundamental groups do not admit any maps of non-zero degree from products of closed manifolds of positive dimension. Particular examples include all manifolds of non-positive sectional curvature of rank one and all irreducible locally symmetric spaces of …
Paper provides lower bounds for eigenvalues on singular Riemannian foliations.
problem Lower bounds for the first non-zero basic eigenvalue on singular Riemannian manifolds.
method Generalized Zhong-Yang and Shi-Yang estimates for singular Riemannian foliations with basic mean curvature.
result Rigidity result when the first basic eigenvalue equals a specific value.
Upper bounds for Steklov eigenvalues on curved submanifolds.
problem Eigenvalue bounds for Steklov problem on submanifolds.
method Reilly-type upper bounds for p-Steklov eigenvalues. result Proved upper bounds for the first non-zero eigenvalue.
We consider regular surfaces M that are given as the zeros of a polynomial function p:R3→R, where the gradient of p vanishes nowhere. We assume that M has non-zero mean curvature and prove that there exist only two examples of such surfaces, namely the sphere and the circular cylinder.
We construct explicit compact supersymmetric solutions with non-zero field strength, non-flat instanton and constant dilaton to the heterotic string equations in dimension five. We present a quadratic condition on the curvature which is necessary and sufficient the heterotic supersymmetry and the anomaly cancellation t…
The paper provides an intrinsic proof of a theorem about Landsberg spaces.
problem Proving Numata's theorem on Landsberg spaces of scalar curvature.
method Intrinsic point of view and coordinate-free proof using Finsler geometry.
result All Landsberg spaces of dimension n≥3 of non-zero scalar curvature are Riemannian spaces of constant curvature. A map between manifolds is an isometry if it's Lipschitz and scalar curvature bounded.
problem Characterizing maps between manifolds based on their scalar curvature and Lipschitz continuity.
method Spectral properties of Dirac operators and index theory for low regularity metrics and bundles.
result A 1-Lipschitz map between manifolds is an isometry if it has bounded scalar curvature.
Study Hamiltonian stationary Lagrangian surfaces in complex space forms.
problem Characterize Lagrangian surfaces with harmonic mean curvature in complex space forms.
method Analyze surfaces with constant and harmonic mean curvature, using second fundamental form parallelism and Gaussian curvature constancy.
result Complete classification of Lagrangian surfaces with harmonic mean curvature and constant Gaussian curvature.
The paper defines subdifferentials on Hadamard manifolds and identifies conditions for Fenchel conjugate equality.
problem Understanding convex analysis on Riemannian manifolds.
method Using Busemann functions to define subdifferentials and investigate Fenchel conjugate equality.
result Identifies conditions for equality in the Fenchel-Young inequality on Hadamard manifolds.
Example surfaces with curvature bounds but no Laplacian eigenvalue lower bound.
problem Bounding curvature does not guarantee positive Laplacian eigenvalues.
method Explicit construction of surfaces with bounded curvature and diameter.
result Found surfaces without positive Laplacian eigenvalue lower bound.
MATLAB toolbox pde2path solves geometric PDEs and finds bifurcations in immersed surfaces.
problem Finding bifurcations in geometric PDEs of immersed surfaces.
method Solving PDEs for surface displacement, updating surface, detecting and localizing bifurcations, and switching branches.
result Symmetry breaking bifurcations in various geometric surfaces.
Study derives a limit functional for Willmore graphs with curvature penalization.
problem Optimizing Willmore graphs with curvature constraints.
method Interpreting penalization as Lagrange multiplier, deriving Γ-limit. result Derives a new limit functional for Willmore graphs.
Channel surfaces are surfaces of revolution, parallel to catenoids or rotational surfaces.
problem Characterizing linear Weingarten surfaces in Euclidean space.
method Demonstrated through explicit parametrizations and deduced existence of complete hyperbolic surfaces.
result Channel linear Weingarten surfaces are surfaces of revolution, parallel to catenoids or rotational surfaces of non-zero constant Gauss curvature.
New criterion for Ricci-flat manifolds with non-vanishing Rosenberg index.
problem Existence of parallel spinors on Ricci-flat manifolds.
method Generalization of existing results to manifolds with non-vanishing Rosenberg index.
result Every closed connected Ricci-flat spin manifold of dimension ≥ 2 with non-vanishing Rosenberg index has special holonomy.
The paper classifies surfaces with constant mean curvature and rotational symmetry.
problem Surfaces with specific geometric properties in Euclidean space.
method Analyzes surfaces parametrized on an annular domain with rotational symmetry.
result Surfaces with constant mean curvature are of rotational symmetry.
The paper proves theorems about constant mean curvature surfaces approaching infinity.
problem Understanding the behavior of constant mean curvature surfaces as boundaries tend to infinity.
method Proves theorems about lamination limits of sequences of compact surfaces with constant mean curvature.
result Generalizes previous results for minimal surfaces to non-zero constant mean curvature.
The study characterizes biharmonic surfaces in specific 3-manifolds.
problem Characterizing biharmonic surfaces in non-Sasakian contact metric 3-manifolds.
method Characterization through scalar curvature and mean curvature.
result Examples of biharmonic submanifolds constructed and determined 3-manifolds with proper biharmonic foliations.
We consider Ricci flow of complete Riemannian manifolds which have bounded non-negative curvature operator, non-zero asymptotic volume ratio and no boundary. We prove scale invariant estimates for these solutions. Using these estimates, we show that there is a limit solution, obtained by scaling down this solution at a…
The paper explores properties of Finsler manifolds with specific curvature conditions.
problem Investigating Finsler metrics with various curvature conditions.
method Analyzing non-Riemannian (α,β)-metrics and compact Finsler manifolds with specific curvature properties. result Compact Finsler manifolds with relatively non-negative stretch curvature are Landsberg metrics.