Solve supercritical Yamabe problem on manifolds with non-umbilic boundary.
problem Solving supercritical Yamabe problem on manifolds with non-umbilic boundary.
method Building blowing-up solutions for a supercritical perturbation of the Yamabe problem.
result Constructed solutions for a supercritical perturbation of the Yamabe problem on manifolds with non-umbilic boundary.
Study on compactness and blow-up of solutions for Yamabe problems on manifolds with non-umbilic boundaries.
problem Compactness and blow-up behavior of solutions to the Yamabe boundary problem on manifolds with non-umbilic boundaries.
method Analysis of stability and blow-up sequences for solutions under perturbations of mean curvature and scalar curvature.
result Existence of a blowing-up sequence of solutions when perturbing the mean curvature from above or below with a function having a large positive maximum.
Study shows non-umbilical qc hypersurfaces in hyper-Kähler manifolds have specific structures.
problem Characterizing non-umbilical quaternionic contact hypersurfaces in hyper-Kähler manifolds.
method Analyzes properties of quaternionic contact hypersurfaces in hyper-Kähler manifolds, focusing on those that are not totally umbilical.
result Non-umbilical qc hypersurfaces in hyper-Kähler manifolds have induced qc structures locally qc homothetic to the standard 3-Sasakian sphere.
We apply conformal flows of metrics restricted to the orthogonal distribution D of a foliation to study the question: Which foliations admit a metric such that the leaves are totally geodesic and the mixed scalar curvature is positive? Our evolution operator includes the integrability tensor of D, and for the case …
Study on surfaces in Heisenberg group with constant mean curvature.
problem Constant mean curvature surfaces in the Heisenberg group.
method Proves surfaces in a neighborhood of non-umbilic points are solutions to a sinh-Gordon equation with a differential constraint.
result Surfaces in Heisenberg group with constant mean curvature described by solutions to sinh-Gordon equation.
For a regular surface in Euclidean space R3, umbilic points are precisely the points where the Gauss and mean curvatures K and H satisfy H2=K; moreover, it is well-known that the only totally umbilic surfaces in R3 are planes and spheres. But for timelike surfaces in Minkowski space $\mat…
Study finds solutions to curvature equation with boundary conditions.
problem Finding solutions to curvature equation with boundary conditions.
method Established local C2 estimates and used blow-up analysis. result Existence of conformal metrics with prescribed curvature and boundary conditions.
The paper solves fractional scalar curvature problems on conformal infinities.
problem Prescribed fractional scalar curvature on conformal infinities.
method Introduced and solved the fractional scalar curvature problem on conformal infinities.
result Existence of smooth solutions to the fractional Yamabe problem in the endpoint case.
Study investigates minimal surfaces in spherical caps, extending previous findings.
problem Characterizing minimal surfaces with free boundaries and capillary conditions in spherical caps.
method Extending previous half-space intersection properties to warped products and capillary minimal surfaces in high codimension.
result Established a dual operation relating free boundary and capillary minimal surfaces.
Compact solutions persist even with linear perturbations of the mean curvature term.
problem Compactness of solutions to the Yamabe problem on manifolds with boundary.
method Linear perturbation of the mean curvature term, proving compactness of solutions.
result Set of solutions remains compact even with negative perturbations.
We study surfaces with one constant principal curvature in Riemannian and Lorentzian three-dimensional space forms. Away from umbilic points they are characterized as one-parameter foliations by curves of constant curvature, each of these curves being centered at a point of a regular curve and contained in its normal p…
Study on hemisphere threshold for Escobar functional on Riemannian manifolds, revealing mass and boundary invariant behaviors.
problem Analyzing the hemisphere threshold for the Escobar functional on compact Riemannian manifolds.
method Near-threshold landscape organization by boundary invariants, exact evaluation of weighted profile moments, Lyapunov-Schmidt correction, and blow-up analysis.
result At threshold, blow-ups concentrate at umbilic points with vanishing mass and gradient, leading to compactness and hemispherical rigidity.
The paper shows caustics by reflection in projective Finsler metrics have at least four cusps.
problem The problem is to understand caustics in projective Finsler metrics.
method The approach is to study Finsler billiards in convex domains with projective metrics and analyze the caustics formed.
result Caustics by reflection in projective Finsler metrics have at least four cusps.
Let X be an asymptotically hyperbolic manifold and M its conformal infinity. This paper is devoted to deduce several existence results of the fractional Yamabe problem on M under various geometric assumptions on X and M: Firstly, we handle when the boundary M has a point at which the mean curvature is negat…
Compactness theorem for fractional Yamabe problem on asymptotically hyperbolic manifolds.
problem Compactness of solutions to the fractional Yamabe problem on conformal infinity.
method Analyzing convergence of scalar curvature and second fundamental form properties.
result Solution set is compact in C2(M) under specific conditions. Compactness proven for specific dimensions of manifolds with boundary conditions.
problem Compactness of scalar-flat conformal metrics on low-dimensional manifolds with boundary.
method Quantitative analysis of a linear equation associated with the boundary Yamabe problem, positive mass theorem, and properties of the trace-free second fundamental form.
result Proven C2-compactness for 4- and 5-manifolds, and a sum of second-order derivatives of the trace-free second fundamental form is non-negative at possible blow-up points for 5-manifolds. Normal forms and invariants for nondegenerate hypersurfaces in C^2.
problem Equivalence problem for nondegenerate real hypersurfaces in C^2.
method Equivariant moving frames and invariant differentiation.
result A single real differential invariant of order 7 generates the entire algebra of differential invariants for nondegenerate real hypersurfaces at singularly umbilic points.