New contractible domains on half-sphere with constant boundary Laplacian eigenfunctions.
problem Existence of contractible domains with specific boundary conditions.
method Local bifurcation argument around geodesic disks, anisotropic Hölder spaces, computer-assisted techniques.
result Existence of nontrivial contractible domains on half-sphere with constant boundary Laplacian eigenfunctions.
Research shows conditional existence of foliations by CMC and Willmore type half-spheres near a boundary point.
problem Conditional existence of foliations by CMC and Willmore type half-spheres near a boundary point.
method Analyzes the geometry of the domain's boundary to determine foliation conditions.
result Conditional foliation is possible but not guaranteed, depending on the domain's geometry.
Study on half spheres solves Nirenberg problem with complex blow-up analysis.
problem Prescribing scalar curvature on half spheres.
method Refined blow-up analysis of finite energy approximated solutions.
result Complex blow-up points and vortex problems reveal new connections.
This paper is devoted to the problem of prescribing the scalar curvature under zero boundary conditions. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results on the standard half sphere.
The paper solves a Nirenberg problem on half spheres, finding multiple blow-ups.
problem Finding conformal metrics with prescribed scalar curvature and zero boundary mean curvature on half spheres.
method Constructing finite energy solutions to a subcritical approximation of the problem on half spheres of dimension \( n \geq 5 \).
result The solutions exhibit multiple blow-up of cluster-type at the same boundary point.
In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the standard four dimensional half sphere. We provide an Euler-Hopf type criterion for a given function to be a scalar curvature to a metric conformal to the standard one. Our proof involves the study of crit…
The paper solves the Nirenberg problem on high-dimensional half spheres with pinching conditions.
problem Finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on half spheres.
method Variational approach with pseudogradient and Morse theory to handle non-compactness.
result Existence results for the Nirenberg problem under various pinching conditions.
Small bubbles sliding on a boundary maintain half-spherical shape.
problem Preserving the shape of small bubbles sliding on a boundary.
method Area-preserving Willmore flow, asymptotic analysis, convergence proof.
result The flow keeps a half-spherical shape for all times.
The paper studies how convex surfaces shrink under mean curvature flow with a free boundary.
problem Mean curvature flow of convex surfaces with a free boundary on convex barriers.
method Introduced a new perturbation argument to establish convexity and pinching estimates.
result The flow contracts a sufficiently convex surface to a point in finite time, asymptotic to a half-sphere.
In this paper, we prove the Hijazi inequality on compact Riemannian spin manifolds under two boundary conditions: the condition associated with a chirality operator and the Riemannian version of the $\MIT$ bag condition. We then show that the limiting-case is characterized as being a half-sphere for the first condition…
The paper studies Pansu spheres in a sub-Riemannian 3-sphere and their area-minimizing properties.
problem The study of Pansu spheres and their area-minimizing properties in a sub-Riemannian 3-sphere.
method Calibration arguments.
result The closed half-spheres of S0 with boundary C0 minimize sub-Riemannian area among compact C1 surfaces with the same boundary. The paper establishes inequalities for p-capacitary functions in flat half-spaces.
problem Understanding p-capacitary functions in asymptotically flat half-spaces. method Establishes monotone quantities and mass-capacity inequalities.
result Sharp inequalities attain equality on a Schwarzschild half-space.
We show that a smooth radially symmetric solution u to the graphic Willmore surface equation is either a constant or the defining function of a half sphere in R3. In particular, radially symmetric entire Willmore graphs in R3 must be flat. When u is a smooth radial solution over a puncture…