arXiv research

New operators and curvatures derived from embedded manifolds.

Paper studies critical points of curvature energies in 4D.

A bound on surface area in Riemannian manifolds without totally geodesic surfaces.

Paper proves a Penrose inequality in extrinsic geometry.

We prove the energy identity and the no neck property for a sequence of smooth extrinsic polyharmonic maps with bounded total energy.

Derives formulas for extrinsic Paneitz operator and $Q$-curvature in general dimensions.

Abstract: Develops universal formulae for Q-curvatures and related operators in conformal geometry.

The goal of the present paper is to investigate the algebraic structure of global conformal invariants of submanifolds. These are defined to be conformally invariant integrals of geometric scalars of the tangent and normal bundle. A famous example of a global conformal invariant is the Willmore energy of a surface. In …

Stability of a new map derived from the equator map is analyzed.

Explicit formulas for extrinsic Paneitz operators and Q-curvatures for totally umbilic hypersurfaces.

We consider static spacetimes whose spatial part admits foliations with the extrinsic curvature tensor K_{ab}=0. There are two complementary cases when the gradient of the lapse function points 1) to the direction of foliation or 2) orthogonally to it. Case 1) gives generalization of metrics like Bertotti-Robinson or N…

The paper examines conditions for the equator map to be minimizing or unstable for higher order energy functionals.

The extrinsic Bonnet-Myers theorem is proven for positive Ricci curvature manifolds.

Novel coarse extrinsic curvature for Riemannian submanifolds.

The paper establishes curvature inequalities and rigidity results for CMC and STCMC surfaces in Riemannian and Lorentzian geometry.

Study fourth-order geometric flow of shape operator for co-dimension one immersions.

The topology and the geometry of a surface play a fundamental role in determining the equilibrium configurations of thin films of liquid crystals. We propose here a theoretical analysis of a recently introduced surface Frank energy, in the case of two-dimensional nematic liquid crystals coating a toroidal particle. Our…

Singular Yamabe problems involve changing sign solutions with interesting geometric properties.

Proves uniqueness of geometric flow in various Riemannian manifolds.

We prove that complete submanifolds, on which the Omori-Yau weak maximum principle for the Hessian holds, with low codimension and bounded by cylinders of small radius must have points rich in large positive extrinsic curvature. The lower the codimension is, the richer such points are. The smaller the radius is, the la…

Discretizations of the mean curvature and extrinsic curvature components are constructed on piecewise flat simplicial manifolds, giving approximations for smooth curvature values in a mostly mesh-independent way. These constructions are given in combinatoric form in terms of the extrinsic hinge angles, the intrinsic st…

Paper proves Hamilton's pinching theorem using mean curvature flow.

The study defines and characterizes extrinsic catenaries in hyperbolic space.

Geodesic spheres are the only quasicomplete surfaces in 3-space-forms.

Conformally invariant functionals on the space of knots are introduced via extrinsic conformal geometry of the knot and integral geometry on the space of spheres. Our functionals are expressed in terms of a complex-valued 2-form which can be considered as the cross-ratio of a pair of infinitesimal segments of the knot.…

We obtain an optimal estimate for the extrinsic curvature of an entire minimal graph in $\H^2\times\R$, $\H^2$ the hyperbolic plane.

The study characterizes constant curvature manifolds using ruled surfaces.

Improving a result of Eschenburg and Kim we give a criterion for semisimplicity of pseudo-Riemannian extrinsic symmetric spaces in terms of the shape operator with respect to the mean curvature vector.

Compactness theory for biharmonic maps on degenerating Einstein manifolds.

In this article, we give the integrability conditions for the existence of an isometric immersion from an orientable simply connected surface having prescribed Gauss map and positive extrinsic curvature into some unimodular Lie groups. In particular, we discuss the case when the Lie group is the euclidean unit sphere $…

Unified study of surfaces using Clifford algebras.

Cylinders in warped product spaces have zero curvature.

We study the topology of (properly) immersed complete minimal surfaces $P^2$ in Hyperbolic and Euclidean spaces which have finite total extrinsic curvature, using some isoperimetric inequalities satisfied by the extrinsic balls in these surfaces, (see \cite{Pa}). We present an alternative and partially unified proof of…

Develops tensor calculus for submanifolds of arbitrary codimension.

In this paper we prove an extrinsic one-sided curvature estimate for disks embedded in $\mathbb{R}^3$ with constant mean curvature which is independent of the value of the constant mean curvature. We apply this extrinsic one-sided curvature estimate in [24] to prove to prove a weak chord arc type result for these disks…

New method controls surface extrinsic diameter for positive scalar curvature metrics.

Researchers define residue families and use them to solve singular Yamabe problems.

In this article we introduce a generalization of the Newton transformation to the case of a system of endomorphisms. We show that it can be used in the context of extrinsic geometry of foliations and distributions yielding new integral formulas containing generalized extrinsic curvatures.

The paper studies essential spectra of submanifolds in Euclidean spaces.

In the paper we prove, that extrinsic curvature does not impose restrictions on the topology of a contact structure, except the obvious ones.

Paper finds convexity in translating solitons for concave flows.

In this article, using the generalized Newton transformation, we define higher order mean curvatures of distributions of arbitrary codimension and we show that they agree with the ones from Brito and Naveira (Ann. Global Anal. Geom. 18, 371-383 (2000)). We also introduce higher order mean curvature vector fields and we…

Study proves inequality for hypersurfaces and shows almost extremals are close to Wulff shape.

The paper establishes curvature inequalities and rigidity results for surfaces in Riemannian and Lorentzian geometry.

Quaternionic reformulation simplifies surface curvature theory.

We derive extrinsic curvature estimates for compact disks embedded in $\mathbb{R}^3$ with nonzero constant mean curvature.

Duality principle for approximation of geometrical objects (also known as Eudoxus exhaustion method) was extended and perfected by Archimedes in his famous tractate "Measurement of circle". The main idea of the approximation method by Archimedes is to construct a sequence of pairs of inscribed and circumscribed polygon…

Following an approach of the second author for conformally invariant variational problems in two dimensions, we show in four dimensions the existence of a conservation law for fourth order systems, which includes both intrinsic and extrinsic biharmonic maps. With the help of this conservation law we prove the continuit…