New curvature defined via graph resistances leads to Ricci flow.
problem Defining curvature on graph edges for analysis.
method Introducing Ricci--Foster curvature based on effective resistances and studying Ricci flow.
result Existence of solutions to Ricci flow on short time intervals, preservation of nonnegative curvature.
The paper introduces a new type of Ricci flow on graphs to study their curvature.
problem Understanding the curvature of graphs and their convergence properties.
method Proposes a weighted Forman and Lin-Lu-Yau Ricci flow on graphs and proves the existence and uniqueness of solutions.
result The normalized curvature flow on trees converges to a constant curvature metric.
Study of mean curvature flows on graphs in warped product manifolds, focusing on behavior at infinity.
problem Behavior of mean curvature flows on graphs in warped product manifolds, especially at infinity.
method Analysis of curve shortening flow and mean curvature flow on geodesic graphs for various warping functions.
result Long-time existence of mean curvature flows and vanishing of curvature and derivatives at infinity.
Flow preserves curvature sharpness on weighted graphs.
problem Curvature flow on weighted graphs.
method Adapting Bakry-Émery calculus for Markovian preservation and analyzing limits.
result Flow limits to curvature sharp weighted graphs.
Two graphs show mean curvature flow differs from heat flow in dimensions n≥2.
problem Comparing mean curvature flow and heat flow on entire graphs.
method Analyzing two specific graphs in dimensions n≥2.
result Mean curvature flow and heat flow behave differently, with oscillation vs. stabilization.
Study of mean curvature flows in product manifolds with graph structures.
problem Understanding mean curvature flows in product manifolds.
method Analysis of mean curvature type flows of graphs in product manifolds \(N imes R\), establishing long-term existence and convergence with specific conditions.
result Established long-term existence and convergence of mean curvature flows with specified conditions.
The paper solves curvature problems on graphs using a special flow.
problem Solving curvature problems on finite graphs.
method Defined the Calabi flow for a specific curvature type and established its global existence and convergence.
result The solution to the Calabi flow exists globally and converges under certain conditions.
Flow on weighted graphs sharpens Bakry-Émery curvature.
problem Sharp curvature in weighted graphs.
method Bakry-Émery curvature flow on mixed weighted graphs.
result Limits of curvature flow are curvature sharp.
The Qk flow on graphs evolves them up to a time T.
problem Evolution of graphs by Qk curvature flow. method Proving evolution of complete non-compact graphs up to time T.
result Complete graphs evolve by Qk curvature up to time T. The Ollivier Ricci flow with prescribed curvature on infinite graphs.
problem Ricci flow with prescribed curvature on infinite graphs.
method Existence and uniqueness of the solution to the Ricci flow.
result Convergence of the Ricci flow for graphs with girth at least 6.
Study investigates the long-term behavior of graphed surfaces under mean curvature flow.
problem Long-term behavior of surfaces under mean curvature flow with boundary conditions.
method Established longtime-existence of mean curvature flow with Dirichlet boundary conditions and analyzed the projection of the surface.
result The moving shadow of the flowing surface is a weak solution for mean curvature flow with Dirichlet boundary conditions.
The study characterizes heat flow and concentration on directed graphs with a lower Ricci curvature bound.
problem Understanding heat flow and concentration on directed graphs with a specific curvature bound.
method Characterization via gradient estimate and transportation inequality for the heat semigroup.
result Concentration of measure inequality for directed graphs with positive Ricci curvature.
Study shows mean curvature flows converge to a geodesic graph over a totally geodesic hypersurface.
problem Mean curvature flows in warped product manifolds with closed hypersurfaces.
method Investigation of mean curvature flows in specific warped product manifolds with conditions on warping function and Ricci curvature.
result Existence and convergence of mean curvature flows for certain initial hypersurfaces.
New equivalence found between two types of curvature on weighted graphs.
problem Equivalence of Lin--Lu--Yau curvature and 1/2-Ollivier curvature on weighted graphs.
method Proof of equivalence up to scaling for p≥1/2. result Threshold 1/2 is sharp, extending earlier results. Graph Ricci flow reveals hidden hierarchies in stock market correlations.
problem Detecting hidden structures in the complex stock market graph.
method Using graph Ricci curvature and flow techniques to analyze the NASDAQ 100 index.
result Algorithm detects hidden hierarchies, community behavior, and clustering in financial markets.
The paper studies Ricci flow on graphs with prescribed curvature.
problem Characterizing weight evolution on graphs with prescribed curvature.
method Ricci flow with Lin-Lu-Yau curvature prescription.
result Ricci flow converges to weights of prescribed curvature under certain conditions.
Paper proves stability of flow in cotangent bundle for special Lagrangian submanifolds.
problem Stability of generalized Lagrangian mean curvature flow in cotangent bundle.
method New derivative estimates to weaken initial conditions and remove curvature constraints.
result Stability of flow near special Lagrangian submanifolds in cotangent bundle.
The paper proves uniqueness of evolving graphs by mean curvature flow under specific conditions.
problem Proving uniqueness of entire graphs evolving by mean curvature flow.
method Analyzes graphs of locally Lipschitz functions and rotationally symmetric solutions, proving uniqueness under uniform lower bounds and proper graphs.
result Uniqueness of entire graphs evolving by mean curvature flow under specified conditions.
Symmetric graphs flow without singularities on their axis.
problem Preserving symmetry in mean curvature flow.
method Weak solution approach, introducing 'vanity', mean curvature flow approximation.
result Singularities occur only on the axis of symmetry.
Estimates prove existence of curvature flow in curved spaces.
problem Mean curvature flow in curved spaces with boundary conditions.
method A priori estimates and existence proof for curvature flow.
result Existence of curvature flow with asymptotic Dirichlet conditions.
Unified piecewise-linear Ricci flows improve community detection.
problem Improving community detection in graph neural networks.
method Proposed piecewise-linear Ricci curvature flows with surgeries.
result Flow consistently outperforms baseline models on real-world datasets.
Flow preserves area, proving a general isoperimetric inequality.
problem Proving a general isoperimetric inequality in warped product surfaces.
method Mean curvature flow in warped product surfaces, preserving area.
result General isoperimetric inequality for radial graphs in warped product surfaces.
Study on mean curvature flow of graphs in higher dimensions.
problem Analyzing the evolution of graphs under mean curvature flow.
method Derives estimates using a new maximum principle for submanifolds, applies to uniformly area decreasing maps.
result Graphicality and area decreasing property are preserved for uniformly area decreasing maps.
We study a Neumann problem related to the evolution of graphs under mean curvature flow in Riemannian manifolds endowed with a Killing vector field. We prove that in a particular case these graphs converge to a bounded minimal graph which contacts the cylinder over the domain orthogonally along its boundary.
This paper classifies complete translating solitons in 3D space.
problem Understanding translating solitons for mean curvature flow.
method Full classification of complete translating graphs in R^3.
result A complete classification of complete translating graphs in R^3.
Modified flow finds constant mean curvature surfaces in hyperbolic space.
problem Existence of smooth complete hypersurfaces of constant mean curvature in hyperbolic space.
method Modified mean curvature flow (MMCF) as a geometric flow tool.
result MMCF starting from radial graphs exists and stays radially graphic for all time.
The aim of this work is studying translating graphs by mean curvature flow in $\Real^3$. We prove non-existence of complete translating graphs over bounded domains in $\Real^2$. Furthermore, we show that there are only three types of complete translating graphs in $\Real^3$; entire graphs, graphs between two vertical p…
Study examines mean curvature flow on graphs of maps between manifolds.
problem Investigating mean curvature flow on graphs of maps between manifolds with bounded geometry.
method Investigates the mean curvature flow of graphs of smooth length-decreasing maps f:RmoN. result Uniform decay estimates for all derivatives of order ≥2 of ft along the flow. Study inverse mean curvature flow on entire graphs, proving finite time existence for certain asymptotic cases.
problem Analyzing the evolution of entire graphs under inverse mean curvature flow.
method Global existence for starshaped graphs, critical case analysis for asymptotically conical graphs.
result Existence of a finite time \( T \) for certain asymptotically conical graphs, convergence to a flat plane as \( t o T \).
Study on Type IIb mean curvature flow solutions and their asymptotic behavior.
problem Understanding the behavior of Type IIb solutions to mean curvature flow.
method Proved existence of Type IIb solutions under certain conditions and studied their asymptotic behavior.
result Longtime solution to mean curvature flow with specific initial data must be Type IIb.
We study graphical mean curvature flow of complete solutions defined on subsets of Euclidean space. We obtain smooth long time existence. The projections of the evolving graphs also solve mean curvature flow. Hence this approach allows to smoothly flow through singularities by studying graphical mean curvature flow wit…
Sharp estimates for mean curvature flow of graphs are shown and examples are given to illustrate why these are sharp. The estimates improves earlier (non-sharp) estimates of Klaus Ecker and Gerhard Huisken.
The paper classifies and constructs translating graphs in 3D and higher dimensions.
problem Classifying and constructing translating graphs in various dimensions.
method Full classification and construction of translating graphs using mathematical analysis.
result Classification and construction of new examples of translating graphs.
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…
We introduce a sub-Riemannian analogue of the Bence-Merriman-Osher diffusion driven algorithm and show that it leads to weak solutions of the horizontal mean curvature flow of graphs over sub-Riemannian Carnot groups. The proof follows the nonlinear semi-group theory approach originally introduced by L. C. Evans in the…
This paper studies mean curvature flows near cylindrical singularities.
problem Understanding the behavior of mean curvature flows near cylindrical singularities.
method Proved the rescaled flow converges to a graph over a cylinder, defined nondegeneracy, and showed properties of nondegenerate singularities.
result Nondegenerate cylindrical singularities are isolated, have a mean convex neighborhood, and are type-I.
Study nonparametric flows with contact angle conditions in Riemannian manifolds.
problem Mean curvature type flows with contact angle constraints.
method Graphical representation, mean curvature speed, admissible function.
result Long time existence and convergence under specific conditions.
Proves existence and uniqueness of Killing graphs with prescribed curvature.
problem Existence and uniqueness of Killing graphs with prescribed curvature.
method Proves existence and uniqueness of Killing graphs with prescribed mean curvature considering non-constant functions.
result Existence and uniqueness of Killing graphs with prescribed curvature.
We consider (smooth) solutions of the mean curvature flow of graphs over bounded domains in a Lie group free up to step two (and not necessarily nilpotent), endowed with a one parameter family of Riemannian metrics $σ_\e$ collapsing to a subRiemannian metric σ0 as $\e\to 0$. We establish Ck,α estimates for this…
Let M be a complete Riemannian manifold which either is compact or has a pole, and let φ be a positive smooth function on M. In the warped product M×φR, we study the flow by the mean curvature of a locally Lipschitz continuous graph on M and prove that the flow exists for all time an…
We establish the longtime existence and convergence results of the mean curvature flow of entire Lagrangian graphs in Pseudo-Euclidean space which is related to Logarithmic gradient flow.
Smooth solutions found for modified mean curvature flow in Riemannian manifolds.
problem Existence of smooth solutions for modified mean curvature flow.
method A priori estimates for modified mean curvature flow in Riemannian manifolds with Killing vector field.
result Existence of smooth, entire, longtime solutions for modified mean curvature flow with smooth initial data.
Study of mean curvature flow in Fuchsian manifolds, proving convergence for certain initial surfaces.
problem Detecting minimal surfaces in hyperbolic manifolds.
method Investigation of mean curvature flow in Fuchsian manifolds, proving existence and convergence for specific initial surfaces.
result Existence and convergence of mean curvature flow for certain initial surfaces in Fuchsian manifolds.
Proves existence of translating solitons in product manifolds.
problem Existence of translating solitons in MimesR. method Proved existence through mean curvature flow analysis.
result Existence of Jenkins-Serrin graphs as translating solitons.
Study shows long-term existence of IMCF on non-compact graphs in hyperbolic space.
problem Long-term existence of IMCF on non-compact graphs in hyperbolic space.
method Investigation of IMCF on bounded graphs over horospheres, use of cutoff functions, and development of a non-compact ODE maximum principle.
result Long time existence of IMCF on non-compact graphs in hyperbolic space.
Constructs graph manifolds with many Anosov flows.
problem Finding graph manifolds supporting multiple Anosov flows.
method Cutting geodesic flows, pulling back to finite covers, and gluing compatible pairs of flows.
result Constructs graph manifolds with at least n Anosov flows for any n.
We consider the inverse mean curvature flow in smooth Riemannian manifolds of the form ([R0,∞)×Sn,gˉ) with metric gˉ=dr2+ϑ2(r)σ and non-positive radial sectional curvature. We prove, that for initial mean-convex graphs over Sn the flow exists for all times and remains a graph…
In this work, we study graphs in $\M^n\times\Real$ that are evolving by the mean curvature flow over a bounded domain on $\M^n$, with prescribed contact angle in the boundary. We prove that solutions converge to translating surfaces in $\M^n\times\Real$. Also, for a Riemannian manifold $\M^2$ with negative Gaussian cur…