Karen Uhlenbeck's compactness theorem for sequences of connections with L2 bounds on curvature applies only to connections on principal bundles with compact structure group. This article states and proves an extension of Uhlenbecks theorem that describes sequences of connections on principal PSL(2;C) bundles over compa…
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48 results for “Principal curvature theorem”
The paper proves a rigidity theorem for specific minimal hypersurfaces in a sphere.
problem Understanding the rigidity of minimal hypersurfaces in spheres with constant curvature.
method Analyzing the properties of principal curvatures and the second fundamental form.
result Closed minimal hypersurfaces with constant nonnegative scalar curvature are isoparametric under certain conditions.
A new proof of the homogeneity of isoparametric hypersurfaces with six simple principal curvatures (Dorfmeister-Neher's theorem) is given in a method applicable to the multiplicity two case.
In higher dimensions, we classify hypersurfaces with constant mean and scalar curvatures.
problem Classifying hypersurfaces with specific curvature conditions in Euclidean spaces.
method Using principal curvature theorem and a formula for the Laplacian of the squared norm of the second fundamental form.
result Characterization of hypersurfaces with constant mean and scalar curvatures in .
Study eigenvalues of Pucci operator on geodesic balls, extending Cheng's theorem.
problem Eigenvalue problem for the Riemannian Pucci operator on geodesic balls.
method Established upper and lower bounds for Pucci eigenvalues based on curvature.
result Cheng's bounds extended to Pucci eigenvalues on geodesic balls.
New equations reveal how cylinder power in progressive lenses depends on geodesic curvature.
problem Current understanding of cylinder power in progressive lenses is incomplete.
method Derived complete compatibility equations for spatially-varying curvature surfaces.
result Cylinder power depends on geodesic curvature, not just principal curvature.
The study characterizes hypersurfaces in spheres with constant scalar curvature.
problem Characterizing hypersurfaces in spheres with constant scalar curvature.
method Combining intrinsic and extrinsic geometry, establishing Takahashi-type theorems, and deriving integral inequalities.
result Characterizes hypersurfaces with specific curvature properties and provides spherical Bernstein theorems.
In a previous work, the authors gave a definition of `front bundles'. Using this, we give a realization theorem for wave fronts in space forms, like as in the fundamental theorem of surface theory. As an application, we investigate the behavior of principal singular curvatures along A_2-singularities of hypersurfaces w…
We investigate geometric aspects of the the Bäcklund transform of principal contact element nets. A Bäcklund transform exists if and only if it the principal contact element net is of constant negative Gaussian curvature (a pseudosphere). We describe an elementary construction of the Bäcklund transform and prove its co…
Rotational symmetry proven for certain self-expanders.
problem Analyzing self-expanders with decaying principal curvatures.
method Proved a Liouville-type theorem and applied it.
result Rotational symmetry for specific self-expanders.
The study classifies biharmonic real hypersurfaces in complex projective spaces.
problem Classifying biharmonic real hypersurfaces in complex projective spaces.
method Analyzing proper biharmonic Hopf and ruled real hypersurfaces with distinct principal curvatures.
result Biharmonic ruled real hypersurfaces in complex projective spaces are minimal.
The paper compares eigenvalues in curved spaces with spherical models.
problem Comparing eigenvalues in curved spaces with spherical models.
method Min-max formulas and pointwise curvature comparisons.
result Eigenvalue comparisons in curved spaces generalize known results.
The paper classifies and explores isoparametric hypersurfaces in pseudo-Riemannian space forms.
problem Investigating isoparametric hypersurfaces in pseudo-Riemannian space forms.
method Petrov's classification theorem and shape operator analysis.
result No isoparametric hypersurfaces of index 2 with complex principal curvatures exist.
The paper studies constant mean curvature hypersurfaces in Finsler manifolds.
problem Understanding geometric properties of hypersurfaces in Finsler manifolds.
method Using volume preserving variation and homothetic navigation.
result Deduced a Heintze-Karcher type inequality and proved an Alexandrov type theorem.
Compactness theorems for SL(2;C) generalizations of the 4-dimensional anti-self dual equationsmath.DG
Uhlenbeck's compactness theorem can be used to analyze sequences of connections with anti-self dual curvature on principal SU(2) bundles over oriented 4-dimensional manifolds. The theorems in this paper give an extension of Uhlenbeck's theorem for sequences of solutions of certain SL(2,C) analogs of the anti-self dual …
The paper updates methods for studying proper Dupin hypersurfaces in Lie sphere geometry.
problem Understanding proper Dupin hypersurfaces in Lie sphere geometry.
method Using moving frames to study proper Dupin hypersurfaces locally.
result Obtained several classification theorems of proper Dupin hypersurfaces.
The paper improves proximity estimates for hypersurfaces with almost constant curvature in space forms.
problem Proximity to a single sphere for hypersurfaces with curvature functions close to a constant.
method Unified approach using the method of moving planes.
result Sharp quantitative estimates of proximity to a single sphere.
Solves an old problem by showing round spheres are the only compact surfaces with specific curvature properties.
problem Finding compact surfaces in Euclidean 3-space with specific curvature properties.
method Representation of solutions to linear elliptic equations with discontinuous coefficients.
result Compact surfaces of genus zero with specific curvature properties are round spheres.
The study examines hypersurfaces close to constant mean curvature and their proximity to spheres.
problem Understanding hypersurfaces close to constant mean curvature and their proximity to spheres.
method Quantitative stability results for hypersurfaces with mean curvature close to a constant.
result Hypersurfaces close to constant mean curvature are closely related to spheres, with quantitative descriptions of proximity.
We prove that any strongly regular Weingarten surface in Euclidean space carries locally geometric principal parameters. The basic theorem states that any strongly regular Weingarten surface is determined up to a motion by its structural functions and the normal curvature function satisfying a geometric differential eq…
The paper extends curvature concepts to surfaces in normed spaces.
problem Extending curvature concepts to surfaces in normed spaces.
method Using Birkhoff orthogonality and surface areas, the paper characterizes Minkowski Gaussian curvature.
result Characterizations and generalizations of classical theorems for curvature in Minkowski spaces.
The Gauss map on translational Riemannian manifolds helps classify hypersurfaces.
problem Classifying hypersurfaces in translational Riemannian manifolds.
method Introduced translational Riemannian manifolds, defined Gauss map, and proved a Gauss-Bonnet theorem.
result Proved properties of hypersurfaces in unit sphere and reobtained a theorem.
Minimal conformally flat hypersurfaces in space forms have specific geometric properties.
problem Characterizing minimal conformally flat hypersurfaces in space forms with constant sectional curvature.
method Analyzing hypersurfaces with three distinct principal curvatures and constant mean curvature in space forms.
result For minimal hypersurfaces, they are either cones over Clifford tori or a one-parameter family of immersions.
The paper classifies and determines properties of specific hypersurfaces in complex hyperbolic quadrics.
problem Analyzing Hopf hypersurfaces with constant principal curvatures in complex hyperbolic quadrics.
method Classification and determination of principal curvatures for hypersurfaces with different numbers of distinct curvatures.
result Classification and determination of principal curvatures for Hopf hypersurfaces with up to four distinct values.
The study finds energy gaps for Yang-Mills fields on Kähler surfaces.
problem Finding energy gaps for Yang-Mills fields on Kähler surfaces.
method Proving an energy gap result for Yang-Mills connections on Kähler surfaces with positive scalar curvature.
result Proves energy gap results for Yang-Mills fields on Kähler surfaces.
The paper classifies isoparametric hypersurfaces in Minkowski spaces.
problem Classifying isoparametric hypersurfaces in Minkowski spaces.
method Introduced isoparametric functions and hypersurfaces in Finsler manifolds, proving conditions for transnormal functions to be isoparametric.
result Hyperplanes, Minkowski hyperspheres, and -Minkowski cylinders are isoparametric hypersurfaces with specific constant principal curvatures.
Beyond almost Fuchsian spacemath.DG
The study classifies weakly almost Fuchsian manifolds and proves geometric properties.
problem Classifying and understanding weakly almost Fuchsian manifolds.
method Geometric analysis and compactification techniques.
result Uniform upper bounds on volume and Hausdorff dimension for limit sets.
In a previous work, we studied isoparametric functions on Riemannian manifolds, especially on exotic spheres. One result there says that, in the family of isoparametric hypersurfaces of a closed Riemannian manifold, there exist at least one minimal isoparametric hypersurface. In this note, we show such minimal isoparam…
Isoparametric hypersurfaces and hypersurfaces with constant principal curvatures in Finsler spacesmath.DG
Study on isoparametric and constant-curvature hypersurfaces in Finsler spaces.
problem Characterizing and constructing hypersurfaces with specific curvature properties in Finsler geometry.
method Analyzing isoparametric and constant-curvature hypersurfaces on Randers manifolds.
result Construction of a conformally flat Randers manifold with nonisoparametric hyperplanes of constant curvatures.
Study on discrete surfaces with constant principal curvature for nanocarbon applications.
problem Understanding discrete geometry properties of nanocarbon materials.
method Developed discrete surface theory on 3-ary oriented trees, defined discrete principal directions, constructed examples of discrete CPC surfaces.
result Construction of discrete constant principal curvature surfaces, including discrete CPC tori.
The paper studies half-lightlike submanifolds in Lorentzian manifolds with specific distributions.
problem Characterizing half-lightlike submanifolds in Lorentzian manifolds with conformal co-screen distributions.
method Using Cartan's formula, the authors classify half-lightlike submanifolds of Lorentzian space forms with constant screen principal curvatures.
result Screen homothetic half-lightlike submanifolds of a Lorentzian space form with a conformal co-screen distribution are locally lightlike triple product manifolds.
The paper proves rigidity theorems for hypersurfaces in spherical space forms.
problem Proving rigidity of hypersurfaces in spherical space forms under certain curvature conditions.
method Topological and homotopical methods, including diffeomorphisms and weak homotopy equivalences.
result The universal cover of the hypersurface is diffeomorphic to the n-sphere and fundamental group bounds.
New convex ancient solutions found for flows by high powers of curvature.
problem Existence of closed convex ancient solutions to curvature flows.
method Proves existence of closed convex ancient solutions with specific curvature flow speeds.
result Existence of non-homothetic convex ancient solutions for flows by high powers of curvature.
Develops SGH bundles and theories for GC manifolds.
problem No specific problem stated; focuses on new bundle theory.
method Introduces SGH bundles, develops cohomology, and establishes theories.
result Established a Chern-Weil theory and Hodge theory for SGH bundles.
In this article, we give a theorem of reduction of the structure group of a principal bundle P with regular structure group G. Then, when G is in the classes of Lie groups defined by T.Robart [13], we define the closed holonomy group of a connection as the minimal closed Lie subgroup of G for which the previous theorem…
Behavior of principal curvatures of frontals near non-front singular points and their applicationmath.DG
Study behavior of curvatures near singular points of frontals.
problem Understanding frontals near singular points.
method Investigate principal curvatures and vectors near singular points of frontals.
result Extend Ribaucour transformations to frontals with singular points.
In this paper we extend Efimov's Theorem by proving that any complete surface in with Gauss curvature bounded above by a negative constant outside a compact set has finite total curvature, finite area and is properly immersed. Moreover, its ends must be asymptotic to half-lines. We also give a partial so…
On hypersurfaces satisfying conditions determined by the Opozda-Verstraelen affine curvature tensormath.DG
Study on hypersurfaces with specific curvature conditions.
problem Characterizing hypersurfaces with certain curvature properties.
method Defined and analyzed the Opozda-Verstraelen affine curvature tensor for hypersurfaces.
result Conditions for pseudosymmetry types of hypersurfaces with specific curvature properties.
Defines discrete differential geometry concepts in homotopy type theory.
problem No existing definition of Euler characteristic for comparison.
method Type families on higher inductive types, simplicial complexes, principal bundles, connections, curvature, vector fields, index.
result Theorem relating total curvature and total index, key to proving Gauss-Bonnet and Poincaré-Hopf theorems.
Classifies surfaces with special curvature properties.
problem Rotational surfaces with specific curvature conditions.
method Classifies surfaces with rotationally symmetric norms and linear curvature relations.
result Rotational surfaces with linearly related curvatures are classified.
On Einstein hypersurfaces of math.DG
The paper studies Einstein hypersurfaces in a specific warped product space.
problem Investigating Einstein hypersurfaces in a warped product space.
method Analyzing the principal curvatures and using multiply warped product structure.
result Hypersurfaces have at most three distinct principal curvatures and are locally multiply warped products.
Study the geometry of a surface formed by extending a Whitney umbrella.
problem Investigate the geometric properties of a specific surface formed by extending a Whitney umbrella.
method Analyze the intersection with the normal plane, geodesic and normal curvatures, Gaussian and mean curvatures.
result Determine the zeros of curvature functions and deduce geometric relationships.
In this paper are determined the principal curvatures and principal curvature lines on canal surfaces which are the envelopes of families of spheres with variable radius and centers moving along a closed regular curve in R^3. By means of a connection of the differential equations for these curvature lines and real Ricc…
Study proves no minimal surfaces can be contained in certain half-spaces or cones.
problem Prohibiting minimal surfaces from certain geometric configurations.
method Analyzes weighted minimal surfaces in with height-dependent weights.
result No proper surfaces can be contained in specific half-spaces or cones.
The paper classifies Hopf hypersurfaces with constant curvatures on complex quadrics.
problem Classifying Hopf hypersurfaces with specific curvature properties.
method Analyzing hypersurfaces on complex quadrics with at most five distinct constant principal curvatures.
result All classified hypersurfaces are open parts of homogeneous examples.
The study solves a problem related to hypersurfaces in space forms and their conformal counterparts.
problem Characterizing hypersurfaces in pseudo-Riemannian space forms with specific curvature properties.
method Extending results for and providing new solutions for .
result Characterization of hypersurfaces with three distinct principal curvatures and their relation to conformally flat hypersurfaces.
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.
Gradients are natural first order differential operators depending on Riemannian metrics. The principal symbols of them are related to the enveloping algebra and higher Casimir elements. We give certain relations in the enveloping algebra, which induce not only identities for higher Casimir elements but also all Bochne…