Anisotropic curvature flow studied for planar networks.
problem Geometric evolution of planar networks under anisotropic curvature.
method Local existence of classical solutions in the presence of multiple smooth anisotropies.
result Discussion of polycrystalline case aspects.
Proves higher regularity for anisotropic inverse mean curvature flow.
problem Higher regularity of solutions to anisotropic inverse mean curvature flow.
method Proves Harnack estimate and constructs smooth solutions from C1 initial sets. result Smooth solutions become smooth outside a compact set.
We study surfaces with constant anisotropic mean curvature which are invariant under a helicoidal motion. For functionals with axially symmetric Wulff shapes, we generalize the recently developed twizzler representation of Perdomo to the anisotropic case and show how all helicoidal constant anisotropic mean curvature s…
The paper proves existence of special surfaces in 3D manifolds with constant curvature.
problem Existence of surfaces with constant anisotropic mean curvature.
method Min-max theory applied to elliptic integrands in 3D Riemannian manifolds.
result Existence of smooth surfaces with at most one singular point.
Anisotropic curvature flow of networks shows unique solutions and behavior under finite time.
problem Existence and behavior of networks under anisotropic curvature flow.
method Existence, uniqueness, and regularity of maximal geometric solutions proven.
result Existence of maximal geometric solutions and behavior under finite time.
Study shows how weak inverse anisotropic mean curvature flow behaves at infinity.
problem Understanding the asymptotic behavior of anisotropic mean curvature flow.
method Established local gradient estimates for anisotropic p-harmonic functions and weak solutions of IAMCF. result Weak IAMCF is asymptotic to the expanding Wulff shape solution at infinity.
We show that for elliptic parametric functionals whose Wulff shape is smooth and has strictly positive curvature, any surface with constant anisotropic mean curvature which is a topological sphere is a rescaling of the Wulff shape.
The study finds surfaces with constant anisotropic mean curvature foliated by circles in Euclidean space.
problem Existence and geometric description of surfaces with constant anisotropic mean curvature.
method Analyzes surfaces with constant anisotropic mean curvature of the Dirichlet energy, proving existence and classifying them.
result Existence and geometric description of surfaces foliated by circles with zero anisotropic mean curvature.
Study on anisotropic curvature flow for noncompact convex hypersurfaces.
problem Anisotropic curvature flow of noncompact convex hypersurfaces.
method Flow of complete noncompact convex hypersurfaces with anisotropy determined by a Wulff shape.
result The flow exists for all positive time for initial conditions.
Given a positive function F on Sn which satisfies a convexity condition, for 1≤r≤n, we define for hypersurfaces in Rn+1 the r-th anisotropic mean curvature function Hr;F, a generalization of the usual r-th mean curvature function. We call a hypersurface is anisotropic mini…
In this paper we consider the evolution of a graph-like hypersurface by anisotropic mean curvature flow, under some restrictions on the anisotropic area integrand. We find interior estimates (in both time and space) on the gradient of such hypersurfaces, depending only on the height of the graph and the anisotropic are…
Torsions, curvatures, structure equations and Bianchi identities for locally anisotropic superspaces (containing as particular cases different supersymmetric extensions and prolongations of Riemann, Finsler, Lagrange and Kaluza--Klein spaces) are investigated.
The paper studies curvature measures and volume-preserving flows on convex bodies.
problem Characterizing and understanding convex bodies through anisotropic curvature measures.
method Developed anisotropic curvature measures, used Minkowski formulas and Heintze-Karcher inequalities, and analyzed volume-preserving flows.
result Characterized Wulff shapes via anisotropic curvature measures and proved convergence of volume-preserving flows.
The paper proves stability of Wulff shapes using anisotropic curvature functionals.
problem Stability of Wulff shapes under anisotropic curvature.
method Estimates distance to Wulff shape using Lp-norm of traceless F-Hessian of a foliating function. result Quantitative stability results for anisotropic inequalities and problems.
We show how to compute tensor derivatives and curvature tensors using affine connections. This allows for all computations to be obtained without using coordinate systems, in a way that parallels the computations appearing in classical Riemannian Geometry. In particular, we obtain Bianchi identities for the curvature t…
New theorem proves convex bodies with specific curvature measures are rescaled Wulff shapes.
problem Characterizing convex bodies based on anisotropic curvature measures.
method Analyzing k-th anisotropic curvature measures and their relation to anisotropic perimeter.
result Arbitrary convex bodies with specific curvature measures are rescaled Wulff shapes.
Graphs with bounded anisotropic mean curvature are regular almost everywhere.
problem Understanding the regularity of graphs with anisotropic mean curvature.
method Proving regularity for m-dimensional Lipschitz graphs with anisotropic mean curvature bounded in Lp. result Graphs with bounded anisotropic mean curvature are regular almost everywhere.
Study on surfaces with constant anisotropic mean curvature in 3D space.
problem Characterizing surfaces with constant anisotropic mean curvature.
method Analyzing uniformly elliptic anisotropic functionals and proving properties of surfaces.
result Characterization of surfaces with constant anisotropic mean curvature.
Paper studies stability of curved surfaces in a half-space.
problem Stability of anisotropic capillary hypersurfaces in a half-space.
method Analyzes weak stability and proves Bernstein-type theorems.
result Compact hypersurfaces are stable if and only if they are a truncated Wulff shape.
We prove short-time existence of φ-regular solutions to the anisotropic and crystalline curvature flow of immersed planar curves.
New method approximates anisotropic curve shortening flow.
problem Approximating anisotropic curve shortening flow.
method Weak formulation and finite element approximation.
result Optimal H1-error bound for approximation. The paper proves regularity for varifolds with bounded anisotropic mean curvature.
problem Regularity of varifolds with bounded anisotropic mean curvature.
method Local anisotropic regularity theorem and touching balls approach.
result Varifolds can be covered by countably many C2-regular submanifolds. The study shows that certain graphs are regular at boundary points.
problem Boundary regularity of anisotropic minimal Lipschitz graphs.
method Proves regularity for graphs with bounded anisotropic mean curvature and atomic energy condition.
result Regularity at boundary points with density bounded above by 1/2 + σ.
Paper proves anisotropic Minkowski inequality and related inequalities.
problem Proving anisotropic Minkowski inequality and related inequalities.
method Utilizes a nonlinear potential theoretic approach.
result Sharp anisotropic Minkowski inequality and related inequalities proved.
Study anisotropic flow for capillary hypersurfaces, proving new inequalities.
problem Anisotropic capillary hypersurfaces and their properties.
method Anisotropic volume-preserving mean curvature flow, new approach for strictly convex initial hypersurfaces.
result Established new Alexandrov-Fenchel inequalities for strictly convex anisotropic capillary hypersurfaces.
We apply the evolution method to present a new proof of the Alexandrov type theorem for constant anisotropic mean curvature hypersurfaces in the Euclidean space Rn+1.
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.
The paper studies a flow of convex hypersurfaces using anisotropic curvature functions.
problem Deforming convex hypersurfaces in Euclidean space.
method Fully nonlinear curvature flow involving k-th elementary symmetric function and support function.
result Long-time existence and convergence of the flow under certain assumptions.
Study on droplet flow on uneven surfaces, proving existence and properties.
problem Understanding droplet movement on irregular surfaces.
method Existence of smooth flow and 1/2-Hölder continuous minimizing movement solutions.
result Properties of minimizing movements including comparison principles and uniform boundedness.
Study anisotropic inverse Gauss curvature flows and solve dual Orlicz Minkowski problems.
problem Solving dual Orlicz Minkowski problems for anisotropic flows.
method Anisotropic inverse Gauss curvature flows and stationary solutions.
result New existence results for dual Orlicz Minkowski problems for smooth measures.
In this paper, we study the anisotropic Minkowski problem. It is a problem of prescribing the anisotropic Gauss-Kronecker curvature for a closed strongly convex hypersurface in Euclidean space as a function on its anisotropic normals in relative or Minkowski geometry. We first formulate such problem to a Monge-Ampére t…
In this note, we give a classification of complete anisotropic isoparametric hypersurfaces, i.e., hypersurfaces with constant anisotropic principal curvatures, in Euclidean spaces, which is in analogue with the classical case for isoparametric hypersurfaces in Euclidean spaces. On the other hand, by an example of local…
The study shows properties of stable anisotropic minimal hypersurfaces in 4D space.
problem Characterizing stable anisotropic minimal hypersurfaces in R4. method Analyzing the intrinsic cubic volume growth and interior volume upper bounds for stable anisotropic minimal hypersurfaces.
result Explicit estimates of constants for stable anisotropic minimal hypersurfaces in R4. Sharp bounds for anisotropic p-capacity of Euclidean compact sets derived using flow methods.
problem Sharp bounds for anisotropic p-capacity of Euclidean compact sets.
method Inverse anisotropic mean curvature flow (IAMCF) and anisotropic Hawking mass.
result Upper bounds for anisotropic p-capacity derived using flow methods.
We introduce the anisotropic tensor calculus, which is a way of handling with tensors that depend on the direction remaining always in the same class. This means that the derivative of an anisotropic tensor is a tensor of the same type. As an application, we show how to define derivations using anisotropic linear conne…
We present a new implementation of anisotropic mean curvature flow for contour recognition. Our procedure couples the mean curvature flow of planar closed smooth curves, with an external field from a potential of point-wise charges. This coupling constrains the motion when the curve matches a picture placed as backgrou…
The paper introduces a new flow to converge to a Wulff shape from a smooth convex hypersurface.
problem Proving Alexandrov-Fenchel inequalities for anisotropic mixed volumes.
method Introducing a fully nonlinear locally constrained anisotropic curvature flow.
result The flow converges smoothly and exponentially to a scaled Wulff shape.
In this paper we investigate the "area blow-up" set of a sequence of smooth co-dimension one manifolds whose first variation with respect to an anisotropic integral is bounded. Following the ideas introduced by White in (J. Differential Geom., 2016), we show that this set has bounded (anisotropic) mean curvature in the…
Given a positive function F on Sn which satisfies a convexity condition, for 1≤r≤n, we define the r-th anisotropic mean curvature function HrF for hypersurfaces in Rn+1 which is a generalization of the usual r-th mean curvature function. We prove that a compact embedded hypersurface…
Anisotropic obstacle problems and Stefan problem studied with evolving surfaces.
problem Anisotropic parabolic obstacle problems and Stefan problem.
method Cahn-Hoffman transform and anisotropic mean curvature flow.
result Optimal regularity of the solution and C1,α-regularity of the evolving free boundary. Paper extends Wente's result to anisotropic capillary surfaces in half-spaces.
problem Extending Wente's result to anisotropic capillary surfaces.
method New Heintze-Karcher inequality and Minkowski formula.
result Anisotropic capillary hypersurfaces in half-spaces are Wulff shapes.
Study solves a generalized Christoffel-Minkowski problem using curvature flow.
problem Generalization of the Lp-Christoffel-Minkowski problem. method Anisotropic curvature flow to derive long-time existence and smooth solutions.
result Existence of smooth solutions for c=1 under certain initial data. This paper extends Euclidean theorems to anisotropic settings for varifolds.
problem Anisotropic mean curvature of codimension-one varifolds.
method Proves perpendicularity and locality of mean curvature for bounded anisotropic mean curvature varifolds.
result Anisotropic mean curvature agrees with the approximate mean curvature on the rectifiable part of the varifold.
An anisotropic surface energy is the integral of an energy density that depends on the normal at each point over the considered surface, and it is a generalization of surface area. The minimizer of such an energy among all closed surfaces enclosing the same volume is unique and it is (up to rescaling) so-called the Wul…
The paper studies a curvature flow on hypersurfaces in R^(n+1).
problem Analyzing the long-term behavior of a specific type of curvature flow.
method Examining a flow defined by a non-homogeneous anisotropic speed function.
result The flow converges to a sphere for star-shaped and k-convex initial hypersurfaces.
In this paper, we show that the inverse anisotropic mean curvature flow in Rn+1, initiating from a star-shaped, strictly F-mean convex hypersurface, exists for all time and after rescaling the flow converges exponentially fast to a rescaled Wulff shape in the C∞ topology. As an application, we p…
We study the long time existence theory for a non local flow associated to a free boundary problem for a trapped non liquid drop. The drop has free boundary components on two horizontal plates and its free energy is anisotropic and axially symmetric. For axially symmetric initial surfaces with sufficiently large volume…
Study anisotropic obstacle problem for minimal surfaces using Cahn-Hoffman transform.
problem Anisotropic obstacle problem for minimal surfaces.
method Cahn-Hoffman transform to convert to isotropic problem with generalized Robin boundary condition.
result Optimal regularity of the solution and C1,1 regularity of the free boundary.