Study non-minimal surfaces in homogeneous 3-manifolds with constant mean curvature.
problem Classify surfaces of constant mean curvature in homogeneous 3-manifolds.
method Investigate screw motions and classify surfaces in E(κ,τ) including space-forms. result Complete classification of non-minimal surfaces of supercritical constant mean curvature.
Study constant mean curvature tubes in homogeneous spaces.
problem Global geometry of constant mean curvature tubes.
method Screw-motion invariants, foliation, numerical isoperimetric profile.
result Foliation result and embeddedness proof.
Classifies surfaces with zero mean curvature in a light cone.
problem Classifying surfaces with zero mean curvature in a light cone.
method Examined geodesics and screw motions, used Weierstrass representations.
result Complete classification of ruled zero mean curvature surfaces.
Proves Arnold-Thom conjectures for motion by mean curvature.
problem Degenerate elliptic equations and motion by mean curvature.
method Analytic behavior of solutions for C^2 solutions.
result First instances of a general principle in degenerate equations.
Lectures on mean curvature flow and its related equations.
problem Singularity formation, nonuniqueness, and topological change in motion by mean curvature.
method Analyzes motion by mean curvature flow and related equations.
result Exploration of singularity formation, nonuniqueness, and topological change.
Study geometric flow driven by curvature and capacity potential.
problem Understanding motion of sets via curvature and capacity potential.
method Local well-posedness and two weak formulations proposed.
result Established local well-posedness and proposed weak formulations.
We consider the motion by mean curvature of an n-dimensional graph over a time-dependent domain in Rn, intersecting Rn at a constant angle. In the general case, we prove local existence for the corresponding quasilinear parabolic equation with a free boundary, and derive a continuation criteri…
The paper analyzes self-similar solutions for mean curvature flow in 3D.
problem Analyzing self-similar solutions for mean curvature flow in R3. method Analysis of self-similar solutions for surfaces of revolution, ruled surfaces, and cylindrical surfaces under homothetic helicoidal motions.
result Characterization and explicit families of exact solutions for cylindrical surfaces.
We describe all possible self-similar motions of immersed hypersurfaces in Euclidean space under the mean curvature flow and derive the corresponding hypersurface equations. Then we present a new two-parameter family of immersed helicoidal surfaces that rotate/translate with constant velocity under the flow. We look at…
Unified treatment of eigenvalue processes using Riemannian geometry.
problem Eigenvalue processes in various settings.
method Riemannian submersion and gradient flow of isospectral orbits.
result Eigenvalue processes are projections of Brownian motion through Riemannian submersions.
New non-canonical flows found via parabolic Allen-Cahn equations.
problem Existence of non-canonical mean curvature flows inside fattening regions.
method Construction of non-canonical flows as limits of parabolic ε-Allen-Cahn solutions.
result First examples of non-outermost, non-canonical integral Brakke motions.
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…
New method uses anisotropic mean curvature flow for contour recognition.
problem Contour recognition in images.
method Coupling anisotropic mean curvature flow with external charges for curve motion.
result Stable numerical approximation for contour recognition.
The Clifford torus is unstable but rigid in mean curvature flow.
problem Stability and rigidity of the Clifford torus in mean curvature flow.
method Analysis of higher order phenomena, including entropy minimisation and infinitesimal deformations.
result The Clifford torus is locally unique as a self-shrinker for mean curvature flow.
Paper proves minimizing movements match smooth droplet flow in 3D.
problem Consistency of minimizing movements with smooth mean curvature flow.
method Proved minimizing movements coincide with smooth droplet flow.
result Minimizing movements and smooth mean curvature flow are consistent in 3D.
Study properties of self-similar solutions in Mean and Inverse Mean Curvature Flows.
problem Analyze geometric and potential theoretic properties of self-similar solutions.
method Analysis of extrinsic distance function and Brownian motion behavior.
result Parabolic MCF-solitons are self-shrinkers and parabolic IMCF-solitons are self-expanders.
Study on helicoidal singular minimal surfaces with specific properties.
problem Characterizing singular minimal surfaces invariant by helicoidal motions.
method Analyzing surfaces with mean curvature defined by a specific formula and studying their invariance under helicoidal motions.
result Helicoidal singular minimal surfaces have a specific geometric configuration.
Paper introduces a modified Allen-Cahn equation for better energy equipartition.
problem Improving energy equipartition in mean curvature flow.
method Developed a modified Allen-Cahn equation (MAC) by time-dependent parameter and Z-transform.
result Improved estimate on discrepancy energy for better equipartition.
The paper studies MCF solutions on the Heisenberg group and finds linear affine motion functions.
problem Investigating mean curvature flow soliton solutions on the Heisenberg group.
method Analyzes solutions generated by isometries and proves motion functions are linear.
result Function describing motion is always a linear affine function.
On any timelike surface with zero mean curvature in the four-dimensional Minkowski space we introduce special geometric (canonical) parameters and prove that the Gauss curvature and the normal curvature of the surface satisfy a system of two natural partial differential equations. Conversely, any two solutions to this …
In this paper we study motion of surfaces of revolution under the mean curvature flow. For an open set of initial conditions close to cylindrical surfaces we show that the solution forms a "neck" which pinches in a finite time at a single point. We also obtain a detailed description of the neck pinching process.
Suppose curves are moving by curvature in a plane, but one embeds the plane in R3 and looks at the plane from an angle. Then circles shrinking to a round point would appear to be ellipses shrinking to an ``elliptical point,'' and the surface energy would appear to be anisotropic as would the mobility. The result of …
Study of most probable paths for anisotropic Brownian motions on manifolds.
problem Characterizing paths of Brownian motions with anisotropic diffusion on manifolds.
method Using stochastic development and fiber bundle of linear frames, the study provides a comprehensive characterization of most probable paths.
result Explicit equations and integration methods for most probable paths on different geometries, including constant curvature surfaces.
The study finds unique and non-trivial surfaces in complex spaces.
problem Determining complete surfaces with parallel mean curvature.
method Explicit determination of surfaces in complex projective and hyperbolic planes.
result Existence and uniqueness of surfaces in positive curvature, and non-trivial surfaces in negative curvature.
We show some computations related to the motion by mean curvature flow of a submanifold inside an ambient Riemannian manifold evolving by Ricci or backward Ricci flow. Special emphasis is given to the possible generalization of Huisken's monotonicity formula and its connection with the validity of some Li--Yau--Hamilto…
Study shows surfaces in Lorentz manifold evolve by translation.
problem Investigating space-like graphs over compact convex domains in Lorentz manifold.
method Non-parametric mean curvature flow with contact angle boundary condition.
result Solutions converge to translation-only motion.
The paper proves the regularity of solutions to a specific differential equation describing physical phenomena.
problem Understanding the regularity of solutions to a curvature-dependent differential equation.
method Weaves together analysis and geometry to prove the optimal regularity of solutions.
result The second derivative of solutions is continuous only in very rigid situations.
Study surfaces with parallel mean curvature in 4D spaces.
problem Characterize surfaces with parallel normalized mean curvature in Euclidean or Minkowski 4-space.
method Introduced special isothermal parameters and described surfaces using invariant functions.
result Surfaces with parallel normalized mean curvature are uniquely determined by three invariant functions.
The entropy of a hypersurface is given by the supremum over all F-functionals with varying centers and scales, and is invariant under rigid motions and dilations. As a consequence of Huisken's monotonicity formula, entropy is non-increasing under mean curvature flow. We show here that a compact mean convex hypersurface…
Ancient solution found in 3D space with specific symmetry properties.
problem Finding ancient solutions with specific symmetry and geometric constraints in 3D space.
method Constructed a compact, convex ancient solution with O(1)imesO(n) symmetry in a slab of width π. result The only compact, convex, O(n)-invariant ancient solution in a slab of width π. Study classifies translating solitons in Minkowski 3-space, revealing singularities.
problem Classifying translating solitons in Minkowski 3-space.
method Introduced the concept of translating solitons on a light-like direction, classified them into graphical and invariant families.
result All time-like examples are incomplete and some have singularities.
Unified geometric framework for Brownian motion on various manifolds.
problem Modeling Brownian motion on complex Riemannian manifolds.
method Constructing stochastic differential equations with noise and drift terms aligned with Laplace-Beltrami operators.
result Geometrically transparent and mathematically consistent foundation for diffusion processes.
Study timelike surfaces with parallel mean curvature in Minkowski 4-space.
problem Existence and uniqueness of timelike surfaces with parallel mean curvature.
method Introduce canonical parameters and prove existence and uniqueness theorem.
result Each timelike surface with parallel mean curvature is determined by three geometric functions.
Study of invariant surfaces in isotropic and pseudo-isotropic geometries.
problem Prescribed curvature for invariant surfaces in singular metrics.
method Analysis of one-parameter subgroups of isotropic rigid motions, computation of fundamental forms and curvatures.
result Generalization of revolution and helicoidal surfaces to singular metrics.
The paper classifies surfaces in the Heisenberg space invariant under specific isometries.
problem Classifying surfaces in the Heisenberg space with specific geometric properties.
method Analyzing surfaces with mean curvature H=⟨N,∂zangle+λ under left-translations, rotations, and helicoidal motions. result Classification of λ-translators invariant under specific isometries. We consider a system of three surfaces, graphs over a bounded domain in R2, intersecting along a time-dependent curve and moving by mean curvature while preserving the pairwise angles at the curve of intersection (equal to 2π/3.) For the corresponding two-dimensional parabolic free boundary problem we pr…
A well known result of Da Rios and Levi-Civita says that a closed planar curve is elastic if and only if it is stationary under the localized induction (or smoke ring) equation, where stationary means that the evolution under the localized induction equation is by rigid motions. We prove an analogous result for surface…
We prove that a primitive harmonic map is equivariant if and only if it admits a holomorphic potential of degree one. We investigate when the equivariant harmonic map is periodic, and as an application discuss constant mean curvature cylinders with screw motion symmetries.
New ancient and eternal solutions found for mean curvature flow from minimal surfaces.
problem Finding new examples of mean curvature flow solutions.
method Constructing embedded ancient and eternal solutions related to unstable minimal hypersurfaces.
result Found nonconvex, non-soliton solutions to mean curvature flow.
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…
New SDEs use G-Brownian motion, extending mean-field models.
problem Extending mean-field models to new types of stochastic processes.
method Introduced G-SDEs with coefficients dependent on current state and solution as random variable. result Validated new SDE framework for complex stochastic systems.
We study the motion of an n-dimensional closed spacelike hypersurface in a Lorentzian manifold in the direction of its past directed normal vector, where the speed equals a positive power p of the mean curvature. We prove that for any p∈(0,1], the flow exists for all time when the Ricci tensor of the ambient s…
Study curves evolving under curvature motion in networks.
problem Existence and behavior of network curves under curvature motion.
method Existence, uniqueness, singularity formation, asymptotic behavior of network curves.
result Understanding of network curves evolving under curvature motion.
We develop a conservation law for constant mean curvature (CMC) surfaces introduced by Korevaar, Kusner and Solomon, and provide a converse, so as to characterize CMC surfaces by a conservation law. We work with `twizzler' construction, which applies a screw-motion to some base curve. We show that, excluding cylinders,…
Study eternal solutions to Allen-Cahn equation on 3-sphere, connecting Clifford tori to equatorial spheres.
problem Understanding eternal solutions to the Allen-Cahn equation on the 3-sphere.
method Realization of Brakke's motion by mean curvature as a singular limit of Allen-Cahn gradient flows, using classifications and rigidity results.
result Construction of eternal integral Brakke flows connecting Clifford tori to equatorial spheres.
Infinite-time blow-up in high-dimensional mean curvature flow.
problem High-dimensional mean curvature flow with exponential asymptotic behavior.
method New zero number argument approach to handle degenerate equations.
result Flow propagates at exponential asymptotic speed, gradients and speeds increase to infinity.
Classifies ancient ovals in higher dimensional mean curvature flow.
problem Classifying ancient ovals in higher dimensional mean curvature flow.
method Spectral parametrization to classify k-ovals.
result Classifies k-ovals in arbitrary dimensions.
This paper studies the large time existence for the motion of closed hypersurfaces in a radially symmetric potential. In physical, this surface can be considered as an electrically charged membrane with a constant charge per area in a radially symmetric potential. The evolution of such surface has been investigated by …