In this paper, we show that the nonexistence of rotationally symmetric harmonic diffeomorphism between the unit disk without the origin and a punctured disc with hyperbolic metric on the target.
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48 results for “Rotationally symmetric surfaces”
Extends Milnor's criterion to biharmonic functions.
problem Deciding surface type for biharmonic functions.
method Generalizes Milnor's criterion to biharmonic functions.
result Characterizes whether a surface is hyperbolic or parabolic for biharmonic functions.
We consider the mean curvature evolution of rotationally symmetric surfaces. Using numerical methods, we detect critical behavior at the threshold of singularity formation resembling the one of gravitational collapse. In particular, the mean curvature simulation of a one-parameter family of initial data reveals the exi…
Upper bound on index of rotationally symmetric self-shrinking tori.
problem Stability of singularities in mean curvature flow.
method Entropy functional and eigenvalue analysis.
result Upper bound on the index of rotationally symmetric self-shrinking tori.
Formula for Heisenberg group surface areas derived.
problem Deriving a formula for surface areas in Heisenberg groups.
method Analogy of Cauchy's surface area formula in Heisenberg groups.
result Formula for p-area of compact hypersurfaces in Heisenberg groups.
Rotationally symmetric solutions persist after mean curvature flow starts from a double cone.
problem Understanding the symmetry of solutions to mean curvature flow.
method Analyzing solutions coming out of a double cone.
result Rotationally symmetric solutions persist.
Area-preserving mean curvature flow of rotationally symmetric hypersurfaces with free boundariesmath.DG
In this paper, we consider the area-preserving mean curvature flow with free Neumann boundaries. We show that for a rotationally symmetric -dimensional hypersurface in between two parallel hyperplanes will converge to a cylinder with the same area under this flow. We use the geometric properties and the m…
We obtain supremum of the k-th normalized Steklov eigenvalues of all rotational symmetric conformal metrics on the cylinder with k>1. The case k=1 for all conformal metrics has been completely solved by Fraser and Schoen. We give geometric description in terms of minimal surfaces for metrics attaining the supremum. We …
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.
The paper classifies hypersurfaces in Heisenberg groups with rotational symmetry.
problem Classifying hypersurfaces in Heisenberg groups with rotational symmetry.
method Fundamental theorems and earlier results in [3] and [4] were used to classify umbilic hypersurfaces and generate curves for hypersurfaces with constant -mean curvature.
result Complete classification of umbilic hypersurfaces and generating curves in Heisenberg groups .
In this paper, we study the Gauss map of a free boundary minimal surface. The main theorem asserts that if components of the Gauss map are eigenfunctions of the Jacobi-Steklov operator, then the surface must be rotationally symmetric.
We study surfaces in TN that are area-stationary with respect to a neutral Kaehler metric constructed on TN from a riemannian metric g on N. We show that holomorphic curves in TN are area-stationary, while lagrangian surfaces that are area-stationary are also holomorphic and hence totally null. However, in general, are…
We show that any minimal torus in which is Alexandrov immersed must be rotationally symmetric. An analogous result holds for surfaces of constant mean curvature.
A Compactness Theorem for Rotationally Symmetric Riemannian Manifolds with Positive Scalar Curvaturemath.DG
Gromov and Sormani conjectured that sequences of compact Riemannian manifolds with nonnegative scalar curvature and area of minimal surfaces bounded below should have subsequences which converge in the intrinsic flat sense to limit spaces which have nonnegative generalized scalar curvature and Euclidean tangent cones a…
New theorems on compactness and finiteness for specific types of self-shrinkers.
problem Characterizing rotationally symmetric self-shrinkers with constraints.
method Compactness and finiteness theorems for self-shrinkers with specific symmetries and constraints.
result Existence of entropy minimizing self-shrinkers diffeomorphic to for each .
The paper classifies helicoidal surfaces with specific curvature functions.
problem Classifying helicoidal surfaces with prescribed mean curvature.
method Phase space analysis for rotationally symmetric -surfaces.
result Classification theorem for even and increasing on .
New Ricci flow solutions found with rotational symmetry and cone-like singularities.
problem Finding Ricci flow solutions with specific symmetry and singularity properties.
method Rotationally symmetric Ricci flow with scaling-invariant curvature bounds, using approximation method.
result Complete Ricci flow solution with cone-like singularity at the origin.
The study explores special surfaces in a normed space.
problem Constant Gaussian and mean curvature surfaces in normed spaces.
method Analyzes rotational surfaces with specific curvature properties.
result Generalizes catenoid, pseudo-sphere, and Delaunay surfaces.
Estimates for -capacities on symmetric manifolds.
problem Estimating relative -capacities on symmetric manifolds.
method Rotationally symmetric manifolds and novel volumetric estimates.
result Sharp weak -embeddings and precise lower bounds of principal -frequencies.
Sharp inequalities found for orbifold metrics.
problem Bounding systolic ratios on rotationally symmetric orbifolds.
method Analyzing spindle orbifolds and Besse metrics.
result Upper bounds on systolic ratios are attained at Besse metrics.
Paper proves March's criterion for transience on symmetric manifolds.
problem Determining transience on rotationally symmetric manifolds.
method Analyzes bounded non-constant harmonic functions and Dirichlet problem at infinity.
result March's criterion is necessary and sufficient for transience.
We investigate existence and stability of rotationally symmetric critical immersions of variational problems of higher order which were considered by Nitsche.
Holomorphic discs converge to maximal surfaces under specific flows.
problem Understanding the evolution of holomorphic discs under mean curvature flow.
method Mean curvature flow with boundary conditions in the space of oriented lines.
result Holomorphic discs converge to Bishop filling by holomorphic discs under certain conditions.
The paper characterizes a helicoid in a cylinder with minimal area and unique boundary conditions.
problem Characterize the helicoid in a cylinder with minimal area and specific boundary conditions.
method Characterizes the helicoid in a solid cylinder from two perspectives: area minimality and boundary conditions.
result The helicoid is the unique minimal surface with the specified boundary conditions.
Researchers classify and describe -translators in Euclidean space.
problem Classifying and describing -translators in Euclidean space.
method Rotationally symmetric and helicoidal motions.
result For each , there is a -translator intersecting orthogonally the rotation axis.
In this paper, we show that an embedded Weingarten surface in S^3 of genus 1 must be rotationally symmetric, provided that certain structure conditions are satisfied. The argument involves an adaptation of our proof of Lawson's Conjecture for minimal tori.
The paper classifies rotationally symmetric extremal Kähler metrics on complex manifolds.
problem Classifying extremal Kähler metrics on complex manifolds.
method Analyzing polynomial zeros in Calabi's extremal equation.
result No invariant complete extremal Kähler metrics on with positive bisectional curvature.
In this paper, we determine the maximally stable, rotationally invariant domains on the catenoids $\cC_a$ (minimal surfaces invariant by rotations) in the Heisenberg group with a left-invariant metric. We show that these catenoids have Morse index at least 3 and we bound the index from above in terms of the parameter $…
We study the stability of the Positive Mass Theorem using the Intrinsic Flat Distance. In particular we consider the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces whose boundaries are either outermost minimal h…
New comparison theorems for rotationally symmetric self-shrinkers help in proving the uniqueness of the Angenent torus.
problem Uniqueness of the Angenent torus in rotationally symmetric self-shrinkers
method Analyzing profile curves and vertical points of rotationally symmetric self-shrinkers
result Proving the existence and monotonicity of horizontal-point trajectories
The paper studies non-integer curvature flows and proves convergence to spheres under specific conditions.
problem Analyzing the convergence of non-integer curvature flows on rotationally symmetric surfaces.
method Spectral theory of singular Sturm-Liouville operators to construct an eigenbasis and prove convergence.
result The flow converges to a round sphere if the focal points coincide at the poles, otherwise to a non-round Hopf sphere.
Researchers set entropy limits for specific types of self-shrinkers.
problem Understanding entropy limits for self-shrinkers with symmetries.
method Derived explicit entropy bounds for two specific classes of self-shrinkers using isoparametric foliations and symmetry analysis.
result Entropy bounds generalized to new classes of self-shrinkers, extending previous findings.
Notes on Frenet-Serret formulas for curves in flat pseudo-hermitian manifolds.
problem Analyzing curves in flat pseudo-hermitian manifolds.
method Deriving Frenet-Serret formulas and applying them to specific conditions.
result Characterizations of curves and classification based on their geometric properties.
We study a second order differential equation corresponding to rotationally symmetric -harmonic maps between certain noncompact manifolds. We show unique continuation and Liouville's type theorems for positive solutions. Asymptotic properties and the existence of bounded positive solutions are investigated.
We study a second order ordinary differential equation corresponding to rotationally symmetric -harmonic maps. We show unique continuation and Liouville's type theorems for positive solutions. We discuss the existence of bounded positive entire solutions. Asymptotic properties of the positive solutions are investiga…
Motivated by the rich theory of harmonic maps from a 2-sphere, we study biharmonic maps from a 2-sphere in this paper. We first derive biharmonic equation for rotationally symmetric maps between rotationally symmetric 2-manifolds. We then apply the equation to obtain a classification of biharmonic maps in a family of r…
In this paper we establish a general form of the isoperimetric inequality for immersed closed curves (possibly non-convex) in the plane under rotational symmetry. As an application we obtain a global existence result for the surface diffusion flow, providing that an initial curve is -close to a multiply covered ci…
Model predicts growth competition on curved surfaces.
problem Growth dynamics of two subsets on Riemannian manifolds.
method Modeling growth rates on spherically symmetric Riemannian manifolds.
result Conditions for bounded or unbounded growth on different manifolds.
We consider a class of overdetermined problems in rotationally symmetric spaces, which reduce to the classical Serrin's overdetermined problem in the case of the Euclidean space. We prove some general integral identities for rotationally symmetric spaces which imply a rigidity result in the case of the round sphere.
The paper classifies 3D complete gradient Yamabe solitons.
problem Classifying nontrivial 3D complete gradient Yamabe solitons.
method Analyzing properties of Yamabe solitons to show rotationally symmetry.
result Nontrivial non-flat 3D complete steady gradient Yamabe solitons are rotationally symmetric.
In this note, using Calabi's method, we construct rotationally symmetric Kahler-Ricci solitons on the total space of direct sum of fixed hermitian line bundle and its projective compactification, where the curvature of hermitian line bundle is Kahler-Einstein. These examples generalize the construction of Koiso, Cao an…
The study proves geometric inequalities for static convex domains in static rotationally symmetric spaces.
problem Proving geometric inequalities for static convex domains in static rotationally symmetric spaces.
method Locally constrained curvature flow in a static rotationally symmetric space , proving graphical solutions and static convexity preservation.
result Proves weighted geometric inequalities for static convex domains close to a slice of .
3-manifold curvature comparison with rotationally symmetric bodies.
problem Comparing scalar curvature of 3-manifolds with rotationally symmetric boundaries.
method Inspired by Gromov, comparing mean curvatures and induced metrics.
result Flatness of 3-manifolds under certain curvature conditions.
New findings on magnetic geodesic flows and periodic motions.
problem Characterizing superintegrable systems in magnetic geodesic flows.
method Analyzing rotationally symmetric magnetic geodesic flows.
result All sufficiently slow motions in a central magnetic field are periodic under specific curvature and homogeneity conditions.
The flow of symmetric spheres converges to a round sphere.
problem The behavior of symmetric hypersurfaces under inverse mean curvature flow.
method Localized parabolic maximum principle approach.
result The flow homothetically converges to a round sphere.
We study noncompact surfaces evolving by mean curvature flow. Without any symmetry assumptions, we prove that any solution that is -close at some time to a standard neck will develop a neckpinch singularity in finite time, will become asymptotically rotationally symmetric in a space-time neighborhood of its singul…
Study on free boundary minimal hypersurfaces in Schwarzschild space, proving zero Morse index for certain hypersurfaces.
problem Analyzing free boundary minimal hypersurfaces in the Riemannian Schwarzschild space.
method Variational methods and geometric analysis.
result Zero Morse index for certain free boundary rotationally symmetric totally geodesic hypersurfaces in the Riemannian Schwarzschild space.
We study surfaces evolving by mean curvature flow (MCF). For an open set of initial data that are -close to round, but without assuming rotational symmetry or positive mean curvature, we show that MCF solutions become singular in finite time by forming neckpinches, and we obtain detailed asymptotics of that singul…