Sharp lower bound found for geodesic ball eigenvalues.
problem Finding the minimum eigenvalue for geodesic balls.
method Applied Li-Schoen's uniform Poincare inequality for non-negative Ricci curvature manifolds.
result Sharp lower bound of the first Dirichlet eigenvalue for geodesic balls.
In hyperbolic space Hn we set a geodesic ball of radius ρ. Consider a k dimensional minimal submanifold passing through the origin of the geodesic ball with boundary lies on the boundary of that geodesic ball. We prove that its area is no less than the totally geodesic k dimensional submanifold passing through…
Geodesic balls with non-negative Ricci curvature have a sharp lower bound on their first Dirichlet eigenvalue.
problem Finding a sharp lower bound for the first Dirichlet eigenvalue of geodesic balls.
method Quantitative explicit inequality linking the width of geodesic balls to the spectral gap.
result A quantitative inequality relating the width of geodesic balls to the spectral gap between the first Dirichlet eigenvalue and its lower bound.
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.
New method to bound Laplacian eigenvalues of geodesic balls.
problem Computing upper bounds for the first eigenvalue of Laplacian on geodesic balls.
method Transforming metric tensor into rotationally symmetric form preserving geodesic sphere areas.
result Upper bound for Laplacian eigenvalues is sharp and computable using geodesic sphere areas.
This paper confirms volumes of geodesic balls can identify 4D space forms.
problem Determining if a 4D manifold is a space form using geodesic ball volumes.
method Tensor calculus and classical theorems, not topological characterizations.
result Similar results for 4D manifold space forms confirmed.
Study proves uniqueness of certain minimal surfaces in spherical and hyperbolic spaces.
problem Proving uniqueness of free boundary minimal annuli in geodesic balls.
method Using Steklov problem frequency and antipodal map invariance.
result Minimal annuli are congruent to a critical rotational annulus.
New method uses relative capacities of geodesic balls to determine scalar curvature.
problem Determining scalar curvature from geodesic ball volumes.
method Using relative capacities of concentric small geodesic balls.
result Scalar curvature is determined by relative capacities of geodesic balls.
Lower bounds for geodesic ball volume in 3D with Ricci curvature constraints.
problem Finding volume bounds in 3D manifolds with Ricci curvature limits.
method Providing lower bounds for geodesic ball volume with upper bounds on Ricci curvature.
result Established lower bounds for geodesic ball volume under Ricci curvature constraints.
Constructs ε-splitting maps for geodesic balls with non-negative Ricci curvature.
problem Constructing ε-splitting maps for geodesic balls with non-negative Ricci curvature.
method Induction and stratified almost Gou-Gu Theorem for finding directional points; error estimates for projections.
result Constructs ε-splitting maps on concentric geodesic balls with uniformly small radius. Study free boundary minimal submanifolds in geodesic balls in hyperbolic and spherical spaces.
problem Characterize free boundary minimal submanifolds in geodesic balls of hyperbolic and spherical spaces.
method Define and analyze functionals related to critical metrics and spectral indices.
result Critical metrics of defined functionals arise from free boundary minimal immersions in geodesic balls of hyperbolic and spherical spaces.
Constructs minimal annuli with free boundary in hyperbolic 3-space.
problem Finding minimal surfaces with boundary in hyperbolic geometry.
method Constructs families of non-rotational minimal annuli with shared symmetry.
result Bifurcates from hyperbolic catenoids, forming a countable collection.
Geodesic balls are isoperimetric in hyperbolic spaces with certain densities.
problem Proving isoperimetric properties in hyperbolic spaces with specific densities.
method Using geodesic balls and radial, strictly log-convex densities.
result Geodesic balls are isoperimetric in real hyperbolic space HRn. Study shows volume constraints lead to isoperimetric constant bounds in specific metric spaces.
problem Understanding isoperimetric constants in metric measure spaces with measure contraction property.
method Proves local isoperimetric inequalities on essentially non-branching MCP(K,N) spaces with volume constraints and geometric conditions.
result Establishes bounds on isoperimetric constants in smaller geodesic balls.
Paper finds critical metrics with pinched curvature are geodesic balls.
problem Identifying critical metrics with specific curvature constraints.
method Proved isometry to geodesic balls in S^n and provided conditions for the gradient of the potential function.
result Critical metrics with pinched curvature are isometric to geodesic balls in S^n.
Two geodesic balls maximize the third Neumann eigenvalue in hyperbolic space.
problem Maximizing the third eigenvalue of the Neumann Laplacian in hyperbolic space.
method Using the disjoint union of two geodesic balls to prove maximality.
result The third eigenvalue is maximal for the union of two geodesic balls.
The paper proves a transformation theorem under a monotone property of almost Euclidean factors of geodesic balls.
problem The non-increasing property of numbers of almost Euclidean factors of geodesic balls.
method Proves a transformation theorem under a non-decreasing property compared to the non-increasing property.
result Shows that for a manifold with nonnegative Ricci curvature, if its universal cover is polar at infinity and the number of almost Euclidean factors is monotone, then its fundamental group is finitely generated and virtually abelian.
We study the eigenvalue problem for the Riemannian Pucci operator on geodesic balls. We establish upper and lower bounds for the principal Pucci eigenvalues depending on the curvature, extending Cheng's eigenvalue comparison theorem for the Laplace-Beltrami operator. For manifolds with bounded sectional curvature, we p…
Study on second Robin eigenvalue for Laplacian on manifolds.
problem Maximizing the second Robin eigenvalue for geodesic balls in nonpositively curved space forms.
method Comparison theorem and maximization analysis for the second Robin eigenvalue.
result Geodesic balls in nonpositively curved space forms maximize the second Robin eigenvalue.
Study on eigenvalue rate of geodesic balls in asymptotically hyperbolic Einstein manifolds.
problem Rate of decrease of the first Dirichlet eigenvalue of geodesic balls.
method Investigation of eigenvalues in asymptotically hyperbolic Einstein manifolds with nonnegative Yamabe type conformal infinity.
result Two-term asymptotic of eigenvalues is the same as in hyperbolic space for nonnegative Yamabe type conformal infinity.
Simply connected 4-manifolds with specific Weyl tensor are geodesic balls in space forms.
problem Characterizing 4-manifolds with harmonic anti-self dual Weyl tensor.
method Proving isometry to geodesic balls in space forms.
result Simply connected critical metrics are geodesic balls in space forms.
We obtain lower bounds for the first Laplacian eigenvalues of geodesic balls of spherically symmetric manifolds. These lower bounds are only C0 dependent on the metric coefficients.
In this paper, we prove a scalar curvature rigidity result for geodesic balls in S^n. This result contrasts sharply with the recent counterexamples to Min-Oo's conjecture for the hemisphere (cf. [5]).
In constant curvatures spaces, there are a lot of characterizations of geodesic balls as optimal domain for shape optimization problems. Although it is natural to expect similar characterizations in rank one symmetric spaces, very few is known in this setting. In this paper we prove that, in a non-compact rank one symm…
Study rigidity of geodesic balls on manifolds with boundary.
problem Rigidity of geodesic balls on manifolds with boundary.
method Combining generalized Reilly formula with Steklov-type boundary value problems to derive integral inequalities.
result Characterizations of geodesic balls in space forms.
We extend to higher dimensions earlier sharp bounds for the area of two dimensional free boundary minimal surfaces contained in a geodesic ball of the round sphere. This follows work of Brendle and Fraser-Schoen in the euclidean case.
We study bifurcation from a branch of trivial solutions of semilinear elliptic Dirichlet boundary value problems on a geodesic ball, whose radius is used as the bifurcation parameter. In the proof of our main theorem we obtain in addition a special case of an index theorem due to S. Smale.
In Euclidean and Hyperbolic space, and the hemisphere in Sn, geodesic balls maximize the gap λ2−λ1 of Dirichlet eigenvalues, amoung domains with fixed λ1. We prove an upper bound on λ2−λ1 for domains in manifolds with certain curvature bounds. The inequality is sharp on geodesic balls in spaceforms.
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.
Proves rigidity of geodesic balls in spheres under certain deformations.
problem Rigidity of geodesic balls in spheres under smooth deformations.
method Real Killing connection and solution of Dirac operator boundary value problem.
result Rigidity result for geodesic balls in spheres fails for hemispheres.
The study finds the optimal metrics for free boundary minimal surfaces in spherical caps.
problem Optimizing metrics for free boundary minimal surfaces in spherical caps.
method Introducing functionals based on eigenvalues of Steklov-type problems and proving maximizers are induced by immersions.
result Maximizing metrics are induced by free boundary minimal immersions in geodesic balls of a round sphere.
The paper compares heat kernels on manifolds with Robin boundary conditions.
problem Comparing heat kernels on manifolds with different boundary conditions.
method Proving comparison theorems for heat kernels on geodesic balls and minimal submanifolds.
result Eigenvalue comparison theorem for the first Robin eigenvalues on minimal submanifolds.
Sharp bound on scalar curvature integral in 3-manifolds.
problem Bounding the integral of scalar curvature on 3-manifolds.
method Geodesic ball analysis with nonnegative Ricci curvature.
result Integral of scalar curvature is bounded by 8πR for large radii. New characterizations for manifolds with boundary rigidity results.
problem Rigidity results for compact gradient Einstein-type manifolds with boundaries.
method Analyzing manifolds with rigidity results and characterizations.
result New topological and geometric characterizations for manifolds with boundaries.
The paper proves a Neumann eigenvalue sum inequality in non-Euclidean space forms.
problem Proving an inequality involving Neumann eigenvalues in non-Euclidean spaces.
method Analyzing space forms with constant curvature and using geodesic balls.
result Proves a conjecture about Neumann eigenvalues in non-Euclidean spaces.
In this paper we provide a pinching condition for the characterization of the totally geodesic disk and the rotational annulus among minimal surfaces with free boundary in geodesic balls of three-dimensional hyperbolic space and hemisphere. The pinching condition involves the length of the second fundamental form, the …
We prove a so called κ non-inflating property for Ricci flow, which provides an upper bound for volume ratio of geodesic balls over Euclidean ones, under an upper bound for scalar curvature. This result can be regarded as the opposite statement of Perelman's κ non-collapsing property for Ricci flow. These two resul…
Study σ2-curvature and volume of compact manifolds, proving conditions for Einstein metrics and geodesic balls.
problem Understanding σ2-curvature and volume in compact manifolds. method Critical point analysis, volume comparison, variational properties, geodesic balls.
result Sufficient and necessary condition for a critical metric to be Einstein, volume comparison results.
New formulas limit minimal submanifolds' area in curved spaces.
problem Bounding minimal submanifolds' area in curved spaces.
method Developed new monotonicity formulae involving energy-like integrals over non-geodesic sets.
result Imply sharp area bounds for minimal submanifolds through a prescribed point.
The paper proves inequalities and growth rates for Schouten solitons.
problem Understanding the properties of Schouten solitons.
method Proving inequalities and growth rates using the norm of the gradient and scalar curvature.
result The potential function of Shrinking Schouten solitons grows linearly.
The paper sets up eigenvalue comparison theorems for specific Laplacians on manifolds.
problem Eigenvalue comparison theorems for Witten-Laplacian and weighted p-Laplacian on manifolds with modified Ricci curvature. method Established Cheng-type eigenvalue comparison theorems for the first Dirichlet eigenvalues of the Witten-Laplacian and weighted p-Laplacian on geodesic balls. result Successfully set up eigenvalue comparison theorems for the Witten-Laplacian and weighted p-Laplacian. The study improves Bochner inequality on Finsler manifolds to derive important inequalities.
problem Improving Bochner inequality on Finsler manifolds to derive new inequalities.
method Using improved Bochner inequality and its integrated form, the study derives a sharp Poincaré-Lichnerowicz inequality, a new proof for logarithmic Sobolev inequality, and an estimate of geodesic ball volumes.
result Derivation of new inequalities and estimates on Finsler manifolds.
Develop an ABP approach to Sobolev and Michael-Simon inequalities beyond Euclidean volume growth.
problem Developing an ABP approach to Sobolev and Michael-Simon inequalities under volume noncollapsing assumptions.
method Using a refinement of Brendle's contact-set argument to derive lower bounds for the volumes of geodesic balls.
result A Michael-Simon type inequality for immersed submanifolds with nonnegative sectional curvature and volume noncollapsing.
Study shows fundamental gap of horoconvex domains in hyperbolic space has no positive lower bound.
problem Understanding the fundamental gap of horoconvex domains in hyperbolic space.
method Analysis of fundamental gap of geodesic balls as radius goes to infinity.
result Product of fundamental gap and square of diameter has no positive lower bound for horoconvex domains.
Upper bound found for first nonzero Steklov eigenvalue.
problem Finding an upper limit for the first nonzero Steklov eigenvalue.
method Analyzing Riemannian manifolds with curvature bounds and comparing domains to geodesic balls.
result An upper bound for the first nonzero Steklov eigenvalue is established.
The study constructs free boundary CMC annuli in spherical and hyperbolic balls.
problem Finding free boundary CMC annuli in spherical and hyperbolic balls.
method Constructing free boundary CMC annuli with constant mean curvature H in geodesic balls of S^3 and H^3.
result Embedded free boundary CMC annuli exist for certain mean curvatures in both spaces.
New functionals defined for free boundary minimal submanifolds in higher dimensions.
problem Characterizing metrics for free boundary minimal submanifolds in geodesic balls.
method Introducing and studying new functionals Θr,i and Ωr,i for higher-dimensional free boundary minimal submanifolds. result Critical metrics for these new functionals are the metrics induced by free boundary minimal immersions.
In this paper we prove the existence of a solution to the Dirichlet problem for harmonic maps into a geodesic ball on which the squared distance function from the origin is strictly convex. This improves a celebrated theorem obtained by S. Hildebrandt, H. Kaul and K. Widman in 1977. In particular no curvature assumptio…