Lectures on PDEs for creating surfaces with specific curvature.
problem Creating surfaces with a specific mean curvature.
method Solving a quasilinear elliptic PDE to find surfaces.
result Graphs with specific mean curvature can be found using PDEs.
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A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
169,341 papers · 148 categories
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48 results for “mean curvature PDE”
Lecture notes on mean curvature flow for beginners.
problem Existence and uniqueness of mean curvature flow solutions.
method Conditional existence and weak-strong uniqueness theory.
result Accessible introduction for graduate students.
New PDE systems generalize Hawking mass monotonicity.
problem Generalizing Hawking mass monotonicity to initial data sets.
method Introduced new systems of PDE on initial data sets .
result Generalized Geroch's monotonicity formula to initial data sets.
A PDE approach to the existence and regularity of surfaces of minimum mean curvature variationmath.DG
Analyzes surfaces minimizing mean curvature variation using PDEs.
problem Finding surfaces of minimum mean curvature variation.
method Develops an analytic theory using partial differential equations.
result Establishes existence and regularity of minimizers.
Study phase transitions with prescribed mean curvature in Riemannian manifolds.
problem Understanding phase transitions with prescribed mean curvature in geometric settings.
method Analyzing solutions to inhomogeneous semilinear elliptic PDEs, establishing bounds and asymptotics.
result Established upper and lower bounds for eigenvalues of phase transition problems.
Paper solves curvature prescription problem on surfaces with boundary.
problem Prescribing Gaussian and geodesic curvatures on compact surfaces with boundary.
method Mean field-type formulation and variational techniques.
result Existence results for positive, zero, and negative Euler characteristics.
Proves uniqueness of capillary disks in 3D domains.
problem Proving uniqueness of capillary disks in three-dimensional domains modeled by elliptic PDEs.
method Using elliptic PDEs and properties of surfaces in 3D domains, generalizing Nitsche's and Hopf's theorems.
result Generalizes Nitsche's result for capillary constant mean curvature disks in the Euclidean ball.
Weighted -Willmore Spheresmath.DG
Spheres minimize weighted curvature on spheres.
problem Minimizing weighted mean curvature on spheres.
method Solving a limit of Euler-Lagrange equations for problems.
result Solutions have at most three mean curvature values.
Study on Dirichlet problem for mean curvature type PDEs in Riemannian manifolds.
problem Existence and asymptotic behavior of solutions to a specific PDE in Riemannian manifolds.
method Three parts: general existence proof, asymptotic analysis on hyperbolic space, and non-existence of singularities.
result Existence of solutions in general domains and non-existence of isolated singularities in asymptotic cases.
We illustrate the flow or wave character of the metrics and curvatures of evolving manifolds, introducing the Riemann flow and the Riemann wave via the bialternate product Riemannian metric. This kind of evolutions are new and very natural to understand certain flow or wave phenomena in the nature as well as the geomet…
This work represents an application of constant mean curvature graphs (as solutions of the mean curvature PDE) to non-linear non-Darcy flows in porous media. It relates time invariant pressure distribution graphs to graphs of constant mean curvature surfaces. This differential geometric interpretation provides an impor…
New examples of solitons found using submersion techniques.
problem Finding new mean curvature solitons on manifolds.
method Riemannian submersion techniques to reduce PDE to ODE.
result New examples of rotators in hyperbolic space.
New approach analyzes ancient solutions and singularities of mean curvature flow.
problem Analyzing ancient solutions and singularities of mean curvature flow locally modeled on a cylinder.
method Introduces PDE-ODI principle to convert parabolic differential equations into systems of ordinary differential inequalities.
result Establishes the uniqueness of the bowl soliton times a Euclidean factor among ancient, cylindrical flows with dominant linear mode.
We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations...etc) in divergence form. This divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for harmonic maps into homogeneous s…
Paper proves short-term existence of fractional mean curvature flow.
problem Existence of solutions for fractional mean curvature flow with capillary boundary conditions.
method Fixed point argument.
result Short time existence of solutions for fractional mean curvature flow.
Sharp growth estimates for warping functions in warped product manifolds.
problem Estimating growth of warping functions in warped product manifolds.
method Applying an average method in PDE to establish sharp inequalities.
result Sharp inequalities between mean curvature and sectional curvatures of the ambient manifold.
Constructs constant spacetime mean curvature surfaces for hyperboloidal initial data sets.
problem Creating a foliation of constant spacetime mean curvature surfaces for asymptotically hyperboloidal initial data sets.
method Long time limit of volume preserving spacetime mean curvature flow starting from a constant mean curvature foliation.
result Obtains a foliation of constant spacetime mean curvature surfaces as the long time limit.
Solves natural PDEs for minimal Lorentz surfaces in 4D spacetime.
problem Natural PDEs for minimal Lorentz surfaces in .
method Weierstrass type representations and canonical coordinates.
result Explicit solution of the system of natural PDEs.
Total variation and mean curvature flows on a Lie group quotient enhance and denoise crossing structures.
problem Preserving crossing curvilinear structures in image enhancement and denoising.
method Lifting images to the homogeneous space , applying PDEs for TVF and MCF, and using locally optimal differential frames.
result Better preservation of bundle boundaries and angular sharpness in fiber orientation densities at crossings compared to data-driven diffusions.
The paper explores geometric influences on PDE solutions on manifolds.
problem Qualitative behavior of solutions to quasilinear PDEs on Riemannian manifolds.
method Investigates strong and weak maximum principles, compact support principles, and Liouville theorems.
result Identifies thresholds involving curvatures or volume growth to guarantee properties under Keller-Osserman conditions.
Study timelike meridian surfaces in Minkowski 4-space with specific properties.
problem Characterize timelike meridian surfaces with special properties in Minkowski 4-space.
method Analyze different classes of timelike meridian surfaces with constant Gauss curvature, mean curvature, and parallel normalized mean curvature vector field.
result Explicit solutions to PDEs describing timelike surfaces with parallel normalized mean curvature vector field.
Rectifiable varifolds with bounded curvature can be covered by smooth surfaces.
problem Understanding the structure of rectifiable varifolds with bounded curvature.
method Using curvature of arbitrary closed sets and viscosity solutions of PDEs.
result The support of rectifiable varifolds can be covered by smooth submanifolds.
Distance between evolving hypersurfaces is a PDE solution.
problem Tracking the distance between evolving hypersurfaces.
method Elliptic and parabolic PDEs, mean curvature flow.
result Local Harnack inequalities for the distance between evolving hypersurfaces.
Surfaces with Parallel Normalized Mean Curvature Vector Field in Euclidean or Minkowski 4-Spacemath.DG
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 paper solves two cases of the asymptotically flat scalar-flat Yamabe problem with boundary.
problem Finding asymptotically flat scalar-flat metrics on manifolds with boundary.
method Solving elliptic PDEs with sub- and supersolutions.
result Existence of conformally equivalent asymptotically flat scalar-flat metrics.
A sharp lower bound on the mean curvature integral with critical power for integral varifoldsmath.DG
Sharp lower bound found for integral varifolds' mean curvature.
problem Finding a sharp lower bound for the mean curvature integral of integral varifolds.
method Developed a new approach using integral varifolds and mean curvature.
result A sharp lower bound on the mean curvature integral with critical power for integral varifolds.
Study on evolving graphs of functions under mean curvature flow in R^n.
problem Proving long-time existence and convergence of special Lagrangian evolution equation.
method Consider the graph of a function on , deform it by mean curvature flow, and analyze under 2-positivity assumption.
result Proves long-time existence and convergence results under 2-positivity assumption, improving previous results.
The paper studies CMC foliations and their conformal aspects on Riemannian manifolds.
problem Understanding and normalizing CMC foliations on conformally compact manifolds.
method Non-linear PDEs and conformal transformations.
result Locally, any slicing can be made into a CMC foliation by conformal changes.
New formulation tackles arbitrage in volatile markets using eigenvalue bounds.
problem Arbitrage opportunities in volatile markets beyond a certain time horizon.
method Formulated as a stochastic optimal control problem, solved via PDE.
result Characterized arbitrage time horizon through PDE solution.
Develops arithmetic PDE geometry concepts like curvature and cohomology.
problem Creating a geometry framework for arithmetic PDEs.
method Introducing arithmetic analogues of Levi-Civita and Chern connections, then developing curvature and characteristic classes.
result Arithmetic analogues of curvature and characteristic classes have been developed.
Solves PDE system for minimal space-like surfaces in Minkowski space-time.
problem Solving the system of natural PDE's for minimal space-like surfaces.
method Using canonical Weierstrass representations, solves the system explicitly.
result Expresses solutions by means of two holomorphic functions.
Rigidity theorem for ideal surfaces with flat boundary conditions.
problem Characterizing surfaces with flat boundary conditions.
method Analyzing a sixth order nonlinear elliptic PDE and flat boundary conditions.
result Surfaces with small second fundamental form and flat boundary conditions are planar.
Study stabilizes translating solitons in hyperbolic space for MCF.
problem Stability of translating solitons in hyperbolic space.
method Developed theory, constructed rotationally invariant translators, used avoidance principle and maximum principle.
result Horospheres are dynamically stable as radial graphical solutions to MCF.
We prove a vertical halfspace theorem for surfaces with constant mean curvature properly immersed in the product space $\h^2\times\re,$ where $\h^2$ is the hyperbolic plane and $\re$ is the set of real numbers. The proof is a geometric application of the classical maximum principle for second order elliptic …
The paper finds a Weierstrass representation for a specific type of Lorentzian minimal surface.
problem Minimal Lorentzian surfaces in with certain curvature conditions.
method Weierstrass representation with respect to isothermal and canonical parameters.
result Explicit solution to the system of natural PDEs for general type surfaces.
The paper studies a flow of surfaces in spacetime with a focus on curvature evolution.
problem Constructing spacetime foliations and center of mass in General Relativity.
method Volume preserving curvature flow with mean curvature speed.
result The flow converges to a constant curvature limit, providing a new foliation method.
Study surfaces with specific curvature properties in 3D space.
problem Classify surfaces with prescribed linear Weingarten curvature.
method Analyze immersed surfaces in with specific curvature relations.
result Classify rotational surfaces under certain curvature conditions.
MATLAB toolbox pde2path solves geometric PDEs and finds bifurcations in immersed surfaces.
problem Finding bifurcations in geometric PDEs of immersed surfaces.
method Solving PDEs for surface displacement, updating surface, detecting and localizing bifurcations, and switching branches.
result Symmetry breaking bifurcations in various geometric surfaces.
Study of critical points for 4D conformally invariant curvature energies.
problem Analyzing critical points of conformally invariant curvature energies in 4 dimensions.
method Using Noether's theorem and divergence-free potentials, generating an algebraic structure, and considering Palais-Smale sequences.
result Improved energy estimates for critical points under small-energy hypotheses.
The paper studies geometric PDEs for flatness on Riemannian manifolds.
problem Understanding flatness in geometric PDEs.
method Study geometric PDEs of connection-flatness, curvature-flatness, Ricci-flatness, scalar curvature-flatness.
result Introduce new Theorems about flatness in Differential Geometry.
Center manifold analysis can be used in order to investigate the stability of the stationary solutions of various PDEs. This can be done by considering the PDE as an ODE between certain Banach spaces and linearising about the stationary solution. Here we investigate the volume preserving mean curvature flow using such …
Method classifies solutions to elliptic problems in disk-like domains.
problem Classifying solutions to overdetermined elliptic problems in topological disks.
method Poincare-Hopf index theorem approach.
result Analogue of Hopf's uniqueness theorem for constant mean curvature spheres in general analytic context.
The central object of study of this thesis is inverse mean curvature vector flow of two-dimensional surfaces in four-dimensional spacetimes. Being a system of forward-backward parabolic PDEs, inverse mean curvature vector flow equation lacks a general existence theory. Our main contribution is proving that there exist …
Proves existence of Lagrangian mean curvature flow solutions.
problem Desingularizing transverse intersection points of immersed Lagrangians.
method Direct PDE approach using manifolds with corners and a-corners.
result Existence of Lagrangian mean curvature flow solutions with stronger convergence.
We consider Legendrian contact structures on odd-dimensional complex analytic manifolds. We are particularly interested in integrable structures, which can be encoded by compatible complete systems of second order PDEs on a scalar function of many independent variables and considered up to point transformations. Using …
The Clifford tori in the 3-sphere are a one-parameter family of flat, two-dimensional, constant mean curvature (CMC) surfaces. This paper demonstrates that new, topologically non-trivial CMC surfaces resembling a pair of neighbouring Clifford tori connected at a sub-lattice consisting of at least two points by small ca…
New deep learning methods solve symmetric PDEs efficiently.
problem Solving nonlinear symmetric PDEs in high dimensions.
method Design of PointNet and DeepSet neural networks.
result DeepSet networks provide more accurate solutions and gradients.
Finite-Time Singularities of Lagrangian Mean Curvature Flow with Quantitatively Precise Dynamicsmath.DG
Constructs finite-time singularities in Lagrangian mean curvature flow with precise dynamics.
problem Finite-time singularities in Lagrangian mean curvature flow.
method Modulation analysis around shrinking cohomogeneity-one special Lagrangian desingularizations.
result Explicit curvature blow-up rate and precise dynamics of singularities.