Explicit formulas for extrinsic Paneitz operators and Q-curvatures for totally umbilic hypersurfaces.
problem Analyzing Q-curvatures and Paneitz operators for hypersurfaces.
method Explicit formulas for extrinsic Paneitz operators and Q-curvatures for totally umbilic hypersurfaces.
result Explicit formulas for the extrinsic Paneitz operators P_4 and extrinsic Q-curvatures for totally umbilic hypersurfaces in any dimension.
The extrinsic Bonnet-Myers theorem is proven for positive Ricci curvature manifolds.
problem Understanding the structure of compact Riemannian manifolds with positive Ricci curvature.
method Establishing the extrinsic Bonnet-Myers theorem and showing almost rigidity for hypersurfaces.
result Proven the extrinsic Bonnet-Myers theorem for positive Ricci curvature manifolds and demonstrated almost rigidity for hypersurfaces.
Piecewise flat extrinsic curvature approximations for simplicial manifolds.
problem Approximating smooth curvature on irregular meshes.
method Combinatorial constructions using hinge angles and dual tessellations.
result Approximations of extrinsic curvature are mostly mesh-independent.
Novel coarse extrinsic curvature for Riemannian submanifolds.
problem Understanding extrinsic curvature of submanifolds.
method Derived from Wasserstein 1-distance between probability measures.
result New insights and approximation of mean curvature from data.
New operators and curvatures derived from embedded manifolds.
problem Finding obstructions and coupling extrinsic operators.
method Explicit computation of extrinsic Paneitz operator and its applications.
result New extrinsically-coupled fourth and sixth order operators.
Proves uniqueness of geometric flow in various Riemannian manifolds.
problem Proving uniqueness of geometric flow in general Riemannian manifolds.
method Two backward uniqueness theorems for extrinsic geometric flow.
result Backward uniqueness of extrinsic geometric flow in general ambient 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…
Even-dimensional compact Riemannian manifolds with Hölder continuous isometric immersions have surfaces of bounded extrinsic curvature.
problem Characterizing surfaces formed by isometric immersions with low regularity.
method Proving that extrinsic curvature equals intrinsic curvature via an integral identity for the Brouwer degree of the Gauss map.
result Extrinsic curvature of codimension one isometric immersions with Hölder continuous derivatives is bounded.
Paper proves Hamilton's pinching theorem using mean curvature flow.
problem Hamilton's pinching theorem in extrinsic geometry.
method Mean curvature flow approach.
result Proof of Hamilton's pinching theorem.
The study defines and characterizes extrinsic catenaries in hyperbolic space.
problem Understanding catenaries in hyperbolic geometry.
method Defined extrinsic catenaries in hyperbolic plane, characterized them, and proved their relation to minimal surfaces.
result Extrinsic catenaries in hyperbolic space are critical points of a potential functional and generating curves of minimal surfaces.
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.
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.
problem Characterizing manifolds of constant curvature using ruled surfaces.
method Investigating ruled surfaces in 3d Riemannian manifolds, finding stiction curve, distribution parameter, and fundamental forms.
result Identifies necessary and sufficient conditions for extrinsically flat surfaces to be ruled and proves manifold properties.
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.
problem Classifying immersed surfaces in various manifolds.
method Using Clifford algebras to construct formalism for immersed bilegendrian surfaces.
result Full classifications of immersed bilegendrian surfaces in the unit tangent bundle of the 3-sphere.
Derives formulas for extrinsic Paneitz operator and Q-curvature in general dimensions.
problem Calculating extrinsic conformal invariants for hypersurfaces in Riemannian manifolds.
method Explicit formulas derived using local conformal invariants and non-trivial local conformal invariant C. result Explicit formulas for extrinsic Paneitz operator and Q-curvature in general dimensions. We study the topology of (properly) immersed complete minimal surfaces P2 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…
Cylinders in warped product spaces have zero curvature.
problem Characterizing cylinders in warped product spaces.
method Proving cylinders have zero extrinsic and intrinsic curvatures.
result Cylinders in M2imesRn have zero curvature. In this paper we prove an extrinsic one-sided curvature estimate for disks embedded in R3 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.
problem Preventing complete metrics with positive scalar curvature on surfaces within manifolds.
method Interior control for extrinsic diameter of surfaces with positive scalar curvature.
result Closed aspherical manifolds cannot have complete metrics with positive scalar curvature when subsets are removed.
Researchers define residue families and use them to solve singular Yamabe problems.
problem Solving singular Yamabe problems on manifolds with boundary.
method Introducing residue families and using them to construct differential operators.
result Residue families can be written as compositions of degenerate Laplacians for approximate solutions of singular Yamabe problems.
The paper studies compactification of Hamiltonian stationary Lagrangian submanifolds with bounded extrinsic curvature and volume.
problem Compactification of Hamiltonian stationary Lagrangian submanifolds with bounded extrinsic curvature and volume.
method Proves convergence of a subsequence of submanifolds to a limit under uniform bounds on volumes and extrinsic curvatures.
result A subsequence of Hamiltonian stationary Lagrangian submanifolds converges to a limit locally uniformly in Ck away from a finite set of points. The paper studies essential spectra of submanifolds in Euclidean spaces.
problem Investigating the essential spectrum of submanifolds under geometric conditions.
method Analyzing submanifolds in Euclidean spaces with various geometric constraints.
result The essential spectrum of a complete non-compact submanifold is [0,+∞) if the second fundamental form satisfies certain Lp norms. 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.
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.
problem Understanding convexity in translating solitons for concave extrinsic flows.
method Analyzes convexity estimates for translating solitons evolving under concave functions in Rn+1. result Establishes convexity estimates for translating solitons of concave extrinsic geometric 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.
problem Proving anisotropic extrinsic radius pinching inequality for hypersurfaces.
method Analyzes anisotropic mean curvatures and studies equality cases.
result Almost extremal hypersurfaces are close to Wulff shape.
Quaternionic reformulation simplifies surface curvature theory.
problem Prescribed extrinsic curvature of surfaces.
method Quaternionic reformulation of Labourie's theory.
result Simpler proofs and higher-dimensional generalization.
We derive extrinsic curvature estimates for compact disks embedded in R3 with nonzero constant mean curvature.
New curvature measures for 4D manifolds with corners defined and related to Gauss-Bonnet.
problem Defining curvature measures for 4D manifolds with corners.
method Defined two new extrinsic curvature quantities, one conformal invariant, and a new conformally invariant operator.
result Gauss-Bonnet theorem reformulated in terms of new curvature measures.
Paper proves a Penrose inequality in extrinsic geometry.
problem Proving a Penrose inequality in extrinsic geometry.
method Analyzing minimal capillary surfaces and their free energy.
result Established an extrinsic Penrose inequality.
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…
The paper characterizes grim hyperplanes for translating solitons in mean curvature flow.
problem Characterizing grim hyperplanes for translating solitons in mean curvature flow.
method Analyzing translating solitons with nonnegative scalar curvature and mean curvature that do not change signs on each end.
result An embedded translating soliton is either a hyperplane or a grim hyperplane if it has nonnegative scalar curvature and mean curvature that do not change signs on each end.
Abstract: Develops universal formulae for Q-curvatures and related operators in conformal geometry.
problem Q-curvatures and related operators in conformal geometry of embedded manifolds with boundary.
method Holographic construction using singular Yamabe problem and minimal hypersurface with boundary.
result Universal formulae for extrinsic Q-curvatures and associated operators.
New CRB derived for curved models using extrinsic geometry.
problem Estimate curved statistical families accurately.
method Vector generalization of CRB with curvature correction using SDP and SOS relaxations.
result Directional curvature correction provides more accurate estimation.
Study relationships between intrinsic and extrinsic invariants of Riemannian almost k-product manifolds.
problem Find a relationship between intrinsic and extrinsic invariants of Riemannian almost k-product manifolds isometrically immersed in another Riemannian manifold.
method Establish an optimal inequality involving mixed scalar curvature and square of mean curvature.
result Optimal inequality that includes mixed scalar curvature and square of mean curvature.
Paper derives Chen's inequality for a specific type of submanifold in a generalized space form.
problem Deriving Chen's inequality for C-totally real submanifolds in a generalized (κ,μ)-space form. method Using intrinsic and extrinsic invariants, the paper derives Chen's inequality involving scalar curvature, sectional curvatures, and squared mean curvature.
result Inequalities between squared mean curvature and Ricci curvature and between squared mean curvature and k-Ricci curvature are obtained. We prove that every complete connected immersed surface with positive extrinsic curvature K in H2×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…
A bound on surface area in Riemannian manifolds without totally geodesic surfaces.
problem Bounding the area of surfaces in Riemannian manifolds without totally geodesic surfaces.
method Using extrinsic curvature energy to bound surface area.
result The area of any complete surface immersed into M is bounded by a multiple of its extrinsic curvature energy. 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.
problem Understanding the geometry of Lagrangian submanifolds.
method Recalling and analyzing the extrinsic principal tangential and normal directions, and their corresponding curvatures for Lagrangian submanifolds in complex Euclidean spaces.
result Established natural relationships between distinguished tangential and normal directions and their curvatures for Lagrangian submanifolds.
We determine the Hausdorff limit-set of the Euclidean hypersurfaces with large λ1 or small extrinsic radius. The result depends on the Lp 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.
problem Characterizing and verifying equivariant embeddings of symmetric Kählerian manifolds.
method Investigation motivated by Cartan and Wallach's theorem on symmetric spaces, focusing on CPn and parallel plurimean curvature. result If an equivariant embedding has parallel plurimean curvature, it is the extrinsically symmetric one.
Study reveals how manifold geometry impacts linear regression solutions.
problem Impact of manifold geometry on linear regression solutions.
method Linear regression applied to manifold-structured data, focusing on extrinsic geometry.
result Linear regression does not have a unique solution on flat manifolds.
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…