arXiv research

Study finds specific complex projective plane hypersurfaces hitting equality in curvature inequality.

In this paper, we obtain a basic Chen's inequality for a C-totally real submanifold in a generalized $(κ,μ)$-contact space forms involving intrinsic invariants, namely the scalar curvature and the sectional curvatures of the submanifold on left hand side and the main extrinsic invariant, namely the squared mean curvatu…

B. Y. Chen establish the relationship between the Ricci curvature and the squared mean curvature for submanifolds of Riemannian space form with arbitrary codimension. In this paper, we generalize the relationship between the Ricci curvature and the squared norm of mean curvature vector for submanifolds of Bochner Kahle…

For submanifolds tangent to the structure vector field in cosymplectic space forms, we establish a basic inequality between the main intrinsic invariants of the submanifold, namely its sectional curvature and scalar curvature on one side; and its main extrinsic invariant, namely squared mean curvature on the other side…

Proves a pinching theorem for self-shrinkers of mean curvature flow.

Study of equifocal hypersurfaces in symmetric spaces and backward mean curvature flow.

The paper studies how submanifolds of a sphere evolve over time.

We consider a complete biharmonic hypersurface with nowhere zero mean curvature vector field $φ:(M^m,g)\rightarrow (S^{m+1},h)$ in a sphere. If the squared norm of the second fundamental form $B$ is bounded from above by m, and $\int_M H^{- p }dv_g<\infty$, for some $0<p<\infty$, then the mean curvature is constant.

We study perturbations of the Allen-Cahn equation and prove the convergence to forced mean curvature flow in the sharp interface limit. We allow for perturbations that are square-integrable with respect to the diffuse surface area measure. We give a suitable generalized formulation for forced mean curvature flow and ap…

In this paper we prove that an embedded and simply connected constant mean curvature surface with curvature large at a point contains a multi-valued graph around that point on the scale of $|A|^2$, where $|A|^2$ is the norm squared of the second fundamental form. This generalizes Colding and Minicozzi's result for mini…

Study on Hawking and Bartnik masses for specific surfaces.

The paper proves conditions for stable constant mean curvature surfaces in specific manifolds.

The paper calculates $S$-curvature for specific Finsler metrics.

Proves curvature bounds for submanifolds in negatively curved spaces.

The Wintgen inequality (1979) is a sharp geometric inequality for surfaces in the 4-dimensional Euclidean space involving the Gauss curvature (intrinsic invariant) and the normal curvature and squared mean curvature (extrinsic invariants), respectively. In the present paper we obtain a Wintgen inequality for statistica…

In this note we investigate the behaviour at finite-time singularities of the mean curvature flow of compact Riemannian submanifolds M^m_t\hookrightarrow (N^{m+n}, h). We show that they are characterized by the blow-up of a trace A = H \cdot II of the square of the second fundamental form.

Classically, isothermic surfaces are characterized as those surfaces which are "divisible into infinitesimal squares by their curvature lines". This characterization is the direct analogue to the definition of discrete isothermic nets. In order to understand the relations between the discrete and the smooth theory bett…

Study on surface configurations with curvature and elasticity.

New gap theorem for hypersurfaces with constant mean curvature in space forms.

We derive, for the square operator of Yau, an analogue of the Omori-Yau maximum principle for the Laplacian. We then apply it to obtain nonexistence results concerning complete spacelike hypersurfaces with constant higher order mean curvature in the Steady State space.

We show that for an isometric immersion of a complete Riemannian manifold into a Riemannian manifold with non-positive curvature, the norm of the mean curvature vector field is square integrable, then it is minimal. This is a partial affirmative answer of the B. Y. Chen's conjecture.

Formula found for surfaces in Sol_3, leading to gap results.

Study curvature inequalities for real hypersurfaces in complex space forms.

The study classifies complete Lagrangian self-shrinkers in 4D space.

Paper proves non-positivity of eigenvalue for Schrödinger operator on hypersurfaces.

In this note we prove that every two-dimensional entire Willmore graph in $R^3$ with square integrable mean curvature is a plane.

We obtain a basic inequality involving the Laplacian of the warping function and the squared mean curvature of any warped product isometrically immersed in a Riemannian manifold without assuming any restriction on the Riemann curvature tensor of the ambient manifold. Applying this general theory, we obtain basic inequa…

Study CR-statistical submanifolds in holomorphic statistical spaces.

The paper studies the blow-up of conformal mean curvature flow in higher codimension.

Paper proves rigidity of certain 2D Lagrangian shapes in 4D space.

Paper establishes inequality for submanifolds in real space forms with semi-symmetric non-metric connection.

In this paper we introduce slant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. We obtain some results on slant Riemannian submersions of a cosymplectic manifolds. We also give examples and inequalities between the scalar curvature and squared mean curvature of fibres of such slant submer…

Study on surfaces minimizing elastic energy with boundary constraints.

We establish some inequalities of Chen's type between certain intrinsic invariants (involving sectional, Ricci and scalar curvatures) and the squared mean curvature of submanifolds tangent to the structure vector fields of a generalized S-space-form and we discuss the equality cases of them. We apply the obtained resul…

B. Y. Chen established sharp inequalities between certain Riemannian invariants and the squared mean curvature for submanifolds in real space form as well as in complex space form. In this paper we generalize Chen inequalities for submanifolds of Bochner Kaehler manifolds. Moreover, we consider CR-warped product subman…

In 1965, T. J. Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in Euclidean three-space is at least 2π^2. We prove this conjecture using the min-max theory of minimal surfaces.

Generalizing a theorem of Huang, Cheng and Wan classified the complete hypersurfaces of $\mathbb R^4$ with non-zero constant mean curvature and constant scalar curvature. In our work, we obtain results of this nature in higher dimensions. In particular, we prove that if a complete hypersurface of $\mathbb R^5$ has cons…

The paper examines spectral properties and rigidity of self-expanding solutions in mean curvature flows.

Paper examines conditions for singular square metrics to have constant curvature.

Estimates mean curvature in warped product spaces, generalizing known results.

The article recovers the Smale conjecture on a Sasakian 3-sphere using Legendrian mean curvature flow.

The paper proves conditions for a 4D minimal surface to be isoparametric.

For constant mean curvature surfaces of class $C^2$ immersed inside Sasakian sub-Riemannian 3-manifolds we obtain a formula for the second derivative of the area which involves horizontal analytical terms, the Webster scalar curvature of the ambient manifold, and the extrinsic shape of the surface. Then we prove classi…

The paper classifies certain types of Lagrangian translators and self-expanders in complex 2-space.

We consider the reduced Allen-Cahn action functional, which appears as the sharp interface limit of the Allen-Cahn action functional and can be understood as a formal action functional for a stochastically perturbed mean curvature flow. For suitable evolutions of generalized hypersurfaces this functional consists of th…

Proves inequalities for convex hypersurfaces in Euclidean space.

In this paper we establish a general inequality involving the Laplacian of the warping functions and the squared mean curvature of any doubly warped product isometrically immersed in a Riemannian manifold. Moreover, we obtain some geometric inequalities for C-totally real doubly warped product submanifolds of generaliz…

The paper studies how submanifolds in Gaussian space behave under mean curvature flow, showing they typically blow up.