arXiv research

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

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

Novel coarse extrinsic curvature for Riemannian submanifolds.

New operators and curvatures derived from embedded manifolds.

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.

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.

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…

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

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.

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.

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

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

The paper studies essential spectra of submanifolds in Euclidean spaces.

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.

Quaternionic reformulation simplifies surface curvature theory.

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

We prove that if $n$ is even, $(M,g)$ is a compact $n$-dimensional Riemannian manifold whose Pfaffian form is a positive multiple of the volume form, and $y\in C^{1,α}(M;\mathbb{R}^{n+1})$ is an isometric immersion with $n/(n+1)< α\leq 1$, then $y(M)$ is a surface of bounded extrinsic curvature. This is proved by showi…

New curvature measures for 4D manifolds with corners defined and related to Gauss-Bonnet.

Paper proves a Penrose inequality in extrinsic geometry.

We obtain upper bounds for the isoperimetric quotients of extrinsic balls of submanifolds in ambient spaces which have a lower bound on their radial sectional curvatures. The submanifolds are themselves only assumed to have lower bounds on the radial part of the mean curvature vector field and on the radial part of the…

For an embedded conformal hypersurface with boundary, we construct critical order local invariants and their canonically associated differential operators. These are obtained holographically in a construction that uses a singular Yamabe problem and a corresponding minimal hypersurface with boundary. They include an ext…

New CRB derived for curved models using extrinsic geometry.

Study relationships between intrinsic and extrinsic invariants of Riemannian almost k-product manifolds.

We prove that every complete connected immersed surface with positive extrinsic curvature $K$ in $H^2\times R$ must be properly embedded, homeomorphic to a sphere or a plane and, in the latter case, study the behavior of the end. Then, we focus our attention on surfaces with positive constant extrinsic curvature ($K-$s…

We study deformations of Riemannian metrics on a given manifold equipped with a codimension-one foliation subject to quantities expressed in terms of its second fundamental form. We prove the local existence and uniqueness theorem and estimate the existence time of solutions for some particular cases. The key step of t…

We develop variation formulas for the quantities of extrinsic geometry for adapted variations of metrics on almost-product (e.g. foliated) Riemannian manifolds, and apply them to study the total mixed scalar curvature of a distribution -- analogue of the classical Einstein-Hilbert action. The mixed scalar curvature ${\…

The study examines principal directions and curvatures of Lagrangian submanifolds.

We determine the Hausdorff limit-set of the Euclidean hypersurfaces with large $λ_1$ or small extrinsic radius. The result depends on the $L^p$ norm of the curvature that is assumed to be bounded a priori, with a critical behaviour for $p$ equal to the dimension minus 1.

The study characterizes and verifies equivariant embeddings of symmetric Kählerian manifolds.

Study reveals how manifold geometry impacts linear regression solutions.

We study the volume of extrinsic balls and the capacity of extrinsic annuli in minimal submanifolds which are properly immersed with controlled radial sectional curvatures into an ambient manifold with a pole. The key results are concerned with the comparison of those volumes and capacities with the corresponding entit…

We study, from the extrinsic point of view, the structure at infinity of open submanifolds isometrically immersed in the real space forms of constant sectional curvature $κ\leq 0$. We shall use the decay of the second fundamental form of the the so-called tamed immersions to obtain a description at infinity of the subm…

We interpret the setting for a Radon transform as a submanifold of the space of generalized functions, and compute its extrinsic curvature: it is the Hessian composed with the Radon transform.

Geometrically refines Cramér-Rao bound using extrinsic manifold curvature.

Constructing solutions to geometric flows with rotational symmetry.