The paper improves proximity estimates for hypersurfaces with almost constant curvature in space forms.
problem Proximity to a single sphere for hypersurfaces with curvature functions close to a constant.
method Unified approach using the method of moving planes.
result Sharp quantitative estimates of proximity to a single sphere.
The paper improves stability estimates for soap bubble theorem in curved domains.
problem Stability estimates for the Soap Bubble Theorem in curved domains.
method Leveraging Gagliardo-Nirenberg-type interpolation inequalities.
result Optimal stability estimates for Lr deviations of mean curvature from being constant. New proof of Bishop's theorem using soap bubbles with singularities.
problem Proving Bishop's volume comparison theorem for manifolds with Ricci curvature.
method Using isoperimetric hypersurfaces (soap bubbles) with singularities.
result Successfully overcame the challenge of singularities to prove the theorem.
The study examines hypersurfaces close to constant mean curvature and their proximity to spheres.
problem Understanding hypersurfaces close to constant mean curvature and their proximity to spheres.
method Quantitative stability results for hypersurfaces with mean curvature close to a constant.
result Hypersurfaces close to constant mean curvature are closely related to spheres, with quantitative descriptions of proximity.
Alexandrov's Soap Bubble theorem dates back to 1958 and states that a compact embedded hypersurface in RN with constant mean curvature must be a sphere. For its proof, A.D. Alexandrov invented his reflection priciple. In 1982, R. Reilly gave an alternative proof, based on integral identities and inequal…
Can you fill R^n with a froth of "soap bubbles" that meet at most n at a time? Not if they have bounded diameter, as follows from Lebesgue's Covering Theorem. We provide some related results and conjectures.
We provide sharp stability estimates for the Alexandrov Soap Bubble Theorem in the hyperbolic space. The closeness to a single sphere is quantified in terms of the dimension, the measure of the hypersurface and the radius of the touching ball condition. As consequence we obtain a new pinching result for hypersurfaces i…
The study proves manifold properties related to positive scalar curvature.
problem Proving the non-existence of metrics with positive scalar curvature on certain manifolds.
method Use of generalized soap bubbles and prescribed-mean-curvature functionals.
result Proves non-existence of metrics with positive scalar curvature on specific manifolds.
Survey on soap bubble partitions and their stability.
problem Characterizing and stabilizing soap bubble partitions.
method Survey and analysis of recent research.
result Recent advancements in multi-bubble isoperimetric minimizers and stability.
Method of moving planes used for proving symmetry in PDEs and geometric analysis.
problem Proving symmetry properties in PDEs and geometric analysis.
method Method of the moving planes for quantitative studies.
result Quantitative approximate symmetry results obtained.
The paper proves rigidity results for Serrin-type problems in manifolds.
problem Proving rigidity for Serrin-type problems in Riemannian manifolds.
method Integral identities and Soap Bubble theorem.
result Rigidity results for annular regions in Einstein manifolds.
Study p-Laplacian equation on Riemannian manifolds with positive Ricci curvature.
problem Overdetermined problem for p-Laplacian equation on compact Riemannian manifolds.
method Introduced a new P-function related to the first nonzero eigenvalue for p-Laplacian, derived integral identities, and applied them to achieve inequalities and the Soap Bubble Theorem.
result Achieved the Heintze-Karcher type inequality and the Soap Bubble Theorem.
In this paper, we show that small spherical soap bubbles in irreducible simply connected symmetric spaces of rank greater than one are constructed from the limits of a certain kind of modified mean curvature flows starting from small spheres in the Euclidean space of dimension equal to the rank of the symmetric space, …
Solves an old problem by showing round spheres are the only compact surfaces with specific curvature properties.
problem Finding compact surfaces in Euclidean 3-space with specific curvature properties.
method Representation of solutions to linear elliptic equations with discontinuous coefficients.
result Compact surfaces of genus zero with specific curvature properties are round spheres.
We prove the following quantitative version of the celebrated Soap Bubble Theorem of Alexandrov. Let S be a C2 closed embedded hypersurface of Rn+1, n≥1, and denote by osc(H) the oscillation of its mean curvature. We prove that there exists a positive ε, depending on n and upper …
For any closed Riemannian manifold X we prove that large isoperimetric regions in X×Rn are of the form X×(Euclidean ball). We prove that if X has non-negative Ricci curvature then the only soap bubbles enclosing a large volume are the products X×(Euclidean sphere). We give an example…
The generalized soap bubble problem seeks the least perimeter way to enclose and separate n given volumes in R^m. We study the possible configurations for perimeter minimizing bubble complexes enclosing more than two regions. We prove that perimeter minimizing planar bubble complexes with equal pressure regions and wit…
The paper solves Serrin-type problems on Riemannian manifolds using new inequalities and identities.
problem Solving Serrin-type problems in Riemannian manifolds.
method Using a Heintze-Karcher inequality, a Soap Bubble result, and a new Pohozaev identity.
result New results on Serrin-type problems in Riemannian manifolds, including rigidity theorems.
Study on Serrin's problem in convex cones with rigidity results and geometric inequalities.
problem Serrin's overdetermined problem in convex cones of Riemannian manifolds.
method Rigidity results, soap bubble theorem, Heintze-Karcher inequality, drift Laplacian analysis.
result Characterization of intersections of geodesic balls with cones in Riemannian manifolds.
Study extends eigenvalue formulas to weighted manifolds and proves global rigidity theorems.
problem Eigenvalue formulas and rigidity theorems for weighted manifolds.
method Extends variational formulae to weighted manifolds, proving global rigidity theorems.
result Global rigidity theorems for critical domains in Gaussian half-space.
Soap bubbles and foams have been extensively studied by scientists, engineers, and mathematicians as models for organisms and materials, with applications ranging from extinguishing fires to mining to baking bread. Here we provide some basic results on the space of planar clusters of n bubbles of fixed topology. We sho…
Study soap bubbles with almost constant higher-order mean curvature, proving unique limits under certain conditions.
problem Understanding the asymptotic behavior of soap bubbles with almost constant higher-order mean curvature.
method Analyzing sequences of bounded C2-domains in Rn+1 converging in volume and perimeter, with k-th mean curvature functions converging in L1. result Finite unions of mutually tangent balls are the only possible limits under natural mean convexity and L∞-control on the mean curvature outside a set of vanishing area. A soap film is actually a thin solid fluid bounded by two surfaces of opposite orientation. It is natural to model the film using one polyhedron for each side. Two problems are to get the polyhedra for both sides to be in the same place without canceling each other out and to model triple junctions without introducing …
The paper examines stable CMC hypersurfaces with boundaries on parallel hyperplanes.
problem Stability of CMC hypersurfaces with free boundaries.
method Analysis and numerical computations.
result Equilibrium hypersurfaces are stable without self-intersection in all dimensions.
Study soap films hanging from frames, proving surface limits and curvature conditions.
problem Understanding soap films with gravity and minimal surfaces.
method Compactness theorem for surfaces with vanishing mean curvature and fixed/converging boundaries.
result Minimal surfaces represent all possible limits of almost-minimal surfaces.
Unified treatment of stability problems in geometry and analysis.
problem Spherical closeness of hypersurfaces under geometric constraints.
method Estimate relating distance to geodesic spheres with norms of traceless Hessian operator.
result Unified treatment of stability problems in geometry and analysis.
Derives equilibrium law for Plateau borders in wet soap films and foams.
problem Equilibrium law for Plateau borders in wet foams and films.
method Rigorous derivation using Gauss' capillarity theory, homotopic spanning condition, and effective compactness theorems.
result Sharp regularity properties of energy minimizers for Plateau borders in wet foams and films.
The paper extends an Alexandrov theorem to Minkowski spacetime.
problem Generalizing Alexandrov's theorem to Minkowski spacetime.
method Adapting conditions for closed codimension-two spacelike submanifolds in Minkowski spacetime.
result A generalized Alexandrov theorem for Minkowski spacetime.
Introduces Alexandrov spaces with curvature below, covering various theorems.
problem Understanding spaces with curvature constraints.
method Explains comparison conditions, globalization, tangent spaces, etc.
result Globalization theorem and other theorems established for Alexandrov spaces.
Generalizes Toponogov theorem to Alexandrov spaces.
problem Estimating curve length in non-Euclidean spaces.
method Generalization of Toponogov theorem.
result Proved the length of a curve in two-dimensional Alexandrov spaces.
Paper proves Toponogov's theorem in Alexandrov geometry.
problem Proving Toponogov's theorem in Alexandrov geometry with lower curvature bound.
method Inspired by Riemannian geometry, uses second variation formula.
result Elementary proof of Toponogov's theorem in Alexandrov geometry.
The paper proves a new theorem in Riemannian geometry and offers a new proof for Toponogov's theorem in Alexandrov geometry.
problem Proving new theorems in Riemannian and Alexandrov geometries.
method Inspired by the proof of the Schur-Toponogov theorem, a new proof of Toponogov's theorem is provided.
result A new theorem in Riemannian geometry and a new proof of Toponogov's theorem in Alexandrov geometry.
In this paper, we introduce a new notion for lower bounds of Ricci curvature on Alexandrov spaces, and extend Cheeger-Gromoll splitting theorem and Cheng's maximal diameter theorem to Alexandrov spaces under this Ricci curvature condition.
Proves quantitative Alexandrov theorem for capillary surfaces.
problem Proving a quantitative version of the Alexandrov theorem for capillary hypersurfaces.
method Quantitative analysis of Montiel-Ros-type argument.
result Generalizes Julin-Niinikoski's result to capillary case.
In this paper we first review the covering space method with constrained BV functions for solving the classical Plateau's problem. Next, we carefully analyze some interesting examples of soap films compatible with the covering space method: in particular, the case of a soap film only partially wetting a space curve, a …
The entropy-degree theorem applies to Alexandrov spaces with curvature constraints.
problem Geometric obstructions and volume bounds in singular spaces.
method Developed new degree theorem for Alexandrov spaces using integral currents.
result Entropy-volume minimization prevents metric singularities in Gromov-Hausdorff limits.
Graph comparison ties to Alexandrov's theorems.
problem Graph comparison conditions on metric spaces.
method Proof of Alexandrov's implications from graph comparisons.
result Complete description of graphs with trivial comparisons.
Paper proves Allard's theorem in Alexandrov spaces.
problem Proving Allard's theorem in non-collapsed Alexandrov spaces.
method Developed an intrinsic proof for Riemannian manifolds, then extended to Alexandrov spaces using approximation theorem.
result Explicit constants for constants in terms of geometric data.
We consider an infinitesimal version of the Bishop-Gromov relative volume comparison condition as generalized notion of Ricci curvature bounded below for Alexandrov spaces. We prove a Laplacian comparison theorem for Alexandrov spaces under the condition. As an application we prove a topological splitting theorem.
Let M be an n-dimensional complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant −κ2. Using the cone total curvature TC(Γ) of a graph Γ which was introduced by Gulliver and Yamada Math. Z. 2006, we prove that the density at any point of a soap film-like…
New integral estimates on substatic manifolds improve Alexandrov Theorem.
problem Improving integral estimates on substatic manifolds.
method Introducing a new vector field with nonnegative divergence.
result Generalization and improvement of integral estimates leading to Alexandrov Theorem.
Alexandrov's theorem asserts that spheres are the only closed embedded constant mean curvature hypersurfaces in space forms. In this paper, we consider Alexandrov's theorem in warped product manifolds and prove a rigidity result in the spirit of Alexandrov's theorem. Our approach generalizes the proofs of Reilly and Ro…
In this paper we provide the first examples of non-flat soap films proven to span tetrahedra. These are members of a continuous two parameter family of soap films with tetrahedral boundaries. Of particular interest is a two parameter subfamily where each spanning soap film has the property that two minimal surfaces mee…
The paper proves a nonlocal version of the Alexandrov Theorem for smooth boundaries.
problem Proving the nonlocal version of the Alexandrov Theorem for sets with smooth boundaries.
method Formulated a necessary and sufficient condition for the theorem to hold, used a specific formula for the tangential derivative of the nonlocal mean curvature, and applied the method of moving planes.
result The only set with smooth boundary and constant nonlocal mean curvature is an Euclidean ball.
Paper extends theorem on covering spaces and Jordan curves.
problem Covering and extending theorems for Alexandrov spaces.
method Introduces proximal homotopic cycles to extend the Mitsuishi-Yamaguchi theorem.
result Extensions of the Mitsuishi-Yamaguchi Good Covering Theorem and Jordan curve theorem.
New definitions weaken conditions for Alexandrov spaces.
problem Weak conditions for Alexandrov spaces with curvature bounds.
method Introducing imaginary comparison and angles, and right/left bounded second derivative.
result New proofs for Doubling and Globalization Theorems.
In this paper we give a new proof for an almost isometry theorem in Alexandrov spaces with curvature bounded below.
What are the possible shapes of various things and why? For instance, when a closed wire or a frame is dipped into a soap solution and is raised up from the solution, the surface spanning the wire is a soap film. What are the possible shapes of soap films and why? Or, for instance, why is DNA like a double spiral stair…