The paper classifies surfaces with constant skew curvature in 3-space forms.
problem Classifying surfaces with constant skew curvature in 3-space forms.
method Variational characterization and flow of binormal vector field.
result Classification of rotational surfaces with constant skew curvature.
Study links Hopf differentials to curvature line flows on time-like CMC surfaces.
problem Understanding the relationship between Hopf differentials and curvature line flows on time-like CMC surfaces.
method Investigation of Hopf differentials and curvature line flows on time-like CMC surfaces in Lorentzian 3-space forms.
result The index of a curvature line flow at an umbilic point depends on the remainder of the Hopf differential's order modulo four.
Let M be a complete Sasakian sub-Riemannian 3-manifold of constant Webster scalar curvature κ. For any point p∈M and any number λ∈R with λ2+κ>0, we show existence of a C2 spherical surface Sλ(p) immersed in M with constant mean curvature λ. Our construction recovers in par…
A surface of constant mean curvature (CMC) equal to H in a sub-Riemannian 3-manifold is strongly stable if it minimizes the functional area+2Hvolume up to second order. In this paper we obtain some criteria ensuring strong stability of surfaces in Sasakian 3-manifolds. We also produce new exampl…
Geodesic spheres are the only quasicomplete surfaces in 3-space-forms.
problem Classifying quasicomplete surfaces in 3-space-forms.
method Using quasicompleteness as a weaker form of completeness, the global geometry of surfaces is determined.
result Geodesic spheres are the only quasicomplete surfaces of constant extrinsic curvature in 3-space-forms.
Three disjoint rays in euclidean 3-space form Borromean rays provided their union is knotted, but the union of any two components is unknotted. We construct infinitely many Borromean rays, uncountably many of which are pairwise inequivalent. We obtain uncountably many Borromean hyperplanes.
In this paper, we study stable constant mean curvature H surfaces in R3. We prove that, in such a surface, the distance from a point to the boundary is less that π/(2H). This upper-bound is optimal and is extended to stable constant mean curvature surfaces in space forms.
The study finds instability conditions for specific surfaces in sub-Riemannian 3-space forms.
problem Stability of volume-preserving area-stationary surfaces with singular curves.
method Proof of stability inequality and sufficient conditions for instability.
result Conditions ensuring instability of specific surfaces in sub-Riemannian 3-space forms.
Study of critical tori for mean curvature energies in Killing submersions.
problem Analyzing surface energies in Killing submersions.
method Symmetry reduction and binormal evolution of critical curves.
result Construction of vertical tori critical for mean curvature energies.
New closed non-CMC biconservative surfaces found in round 3-sphere.
problem Existence of closed biconservative surfaces in space forms.
method Characterization of profile curves and proof of existence using curvature energy.
result Existence of a discrete family of closed, non-CMC biconservative surfaces in S3(ρ). Study of f-biharmonic hypersurfaces in conformally flat spaces.
problem Characterize f-biharmonic hypersurfaces in conformally flat spaces. method Analyze f-biharmonicity of totally umbilical hypersurfaces in various contexts. result Properties of f-biharmonic hypersurfaces in nonpositively curved manifolds. The paper studies f-biharmonic maps and submersions in space forms.
problem Characterizing f-biharmonic maps and submersions in space forms. method Analyzing f-biharmonic curves, developing classifications, and using integrability data. result Proper f-biharmonic developable surfaces exist only in the case of cylinders.