arXiv research

Fractional combinatorial flow improves surface conformal structures.

The paper solves fractional combinatorial flows for prescribed hyperbolic bordered surfaces.

The paper introduces combinatorial Calabi flows to find hyperbolic metrics on surfaces with boundary.

Paper introduces new flows to find circle packings with specific curvature.

New method finds metrics on surfaces with prescribed curvatures using circle packings and surgery.

A new method uses heat diffusion to efficiently solve combinatorial optimization problems.

Topology of the Generic Hamiltonian Dynamical Systems on the Riemann Surfaces given by the real part of the generic holomorphic 1-forms, is studied. Our approach is based on the notion of Transversal Canonical Basis of Cycles (TCB). This approach allows us to present a convenient combinatorial model of the whole topolo…

The paper develops algorithms for finding metrics with prescribed combinatorial curvature on polyhedral surfaces.

Combinatorial Ricci flow finds hyperbolic metrics on 3-manifolds.

For triangulated surfaces, we introduce the combinatorial Calabi flow which is an analogue of smooth Calabi flow. We prove that the solution of combinatorial Calabi flow exists for all time. Moreover, the solution converges if and only if Thurston's circle packing exists. As a consequence, combinatorial Calabi flow pro…

The paper introduces combinatorial curvature and flow for polyhedral surfaces, proving rigidity and solving the Yamabe problem.

We study a fractional conformal curvature flow on the standard unit sphere and prove a perturbation result of the fractional Nirenberg problem with fractional exponent $σ\in (1/2,1)$. This extends the result of Chen-Xu (Invent. Math. 187, no. 2, 395-506, 2012) for the scalar curvature flow on the standard unit sphere.

The paper studies deformation of discrete conformal structures on surfaces using combinatorial curvature flows.

We establish short-time existence of the smooth solution to the fractional mean curvature flow when the initial set is bounded and C^{1,1}-regular. We provide the same result also for the volume preserving fractional mean curvature flow.

This paper studies the combinatorial Yamabe flow on hyperbolic surfaces with boundary. It is proved by applying a variational principle that the length of boundary components is uniquely determined by the combinatorial conformal factor. The combinatorial Yamabe flow is a gradient flow of a concave function. The long ti…

Methods from the geometry of nonholonomic manifolds and Lagrange-Finsler spaces are applied in fractional calculus with Caputo derivatives and for elaborating models of fractional gravity and fractional Lagrange mechanics. The geometric data for such models are encoded into (fractional) bi-Hamiltonian structures and as…

We formulate the fractional Ricci flow theory for (pseudo) Riemannian geometries enabled with nonholonomic distributions defining fractional integro-differential structures, for non-integer dimensions. There are constructed fractional analogs of Perelman's functionals and derived the corresponding fractional evolution …

Study infinite combinatorial Ricci flow on spherical surfaces.

Geodesic flows on surfaces have specific fractional-linear integrals related to constant cross-ratios.

The paper establishes a discrete uniformization theorem for surfaces with piecewise hyperbolic metrics.

Paper proves short-term existence of fractional mean curvature flow.

Paper proves Luo's conjecture for 3D triangulated manifolds.

The paper finds hyperbolic metrics on surfaces with boundary using combinatorial curvature flows.

Study connects taffy pulling, fractions, and rational tangles.

The paper studies the combinatorial p-th Calabi flow for finite and infinite circle patterns.

The paper extends circle pattern flows to hyperbolic and Euclidean geometry.

Computing uniformization maps for surfaces has been a challenging problem and has many practical applications. In this paper, we provide a theoretically rigorous algorithm to compute such maps via combinatorial Calabi flow for vertex scaling of polyhedral metrics on surfaces, which is an analogue of the combinatorial Y…

Paper resolves spherical curvature flow problem.

For triangulated surfaces locally embedded in the standard hyperbolic space, we introduce combinatorial Calabi flow as the negative gradient flow of combinatorial Calabi energy. We prove that the flow produces solutions which converge to ZCCP-metric (zero curvature circle packing metric) if the initial energy is small …

For triangulated surfaces and any $p>1$, we introduce the combinatorial $p$-th Calabi flow which precisely equals the combinatorial Calabi flows first introduced in H. Ge's thesis when $p=2$. The difficulties for the generalizations come from the nonlinearity of the $p$-th flow equation when $p\neq 2$. Adopting differe…

In this paper, we introduce two discrete curvature flows, which are called $α$-flows on two and three dimensional triangulated manifolds. For triangulated surface $M$, we introduce a new normalization of combinatorial Ricci flow (first introduced by Bennett Chow and Feng Luo \cite{CL1}), aiming at evolving $α$ order di…

New method finds ideal circle patterns on spheres.

New discrete conformal structures on surfaces with boundary, proving global rigidity and constructing hyperbolic metrics.

We investigate the combinatorial Ricci flow on a surface of nonpositive Euler characteristic when the necessary and sufficient condition for the convergence of the combinatorial Ricci flow is not valid. This observation addresses one of questions raised by B. Chow and F. Luo.

We introduce a combinatorial curvature flow for PL metrics on compact triangulated 3-manifolds with boundary consisting of surfaces of negative Euler characteristic. The flow tends to find the complete hyperbolic metric with totally geodesic boundary on a manifold. Some of the basic properties of the combinatorial flow…

In this paper, we introduce a new combinatorial curvature on triangulated surfaces with inversive distance circle packing metrics. Then we prove that this combinatorial curvature has global rigidity. To study the Yamabe problem of the new curvature, we introduce a combinatorial Ricci flow, along which the curvature evo…

Study of combinatorial Calabi flow on ideal circle patterns.

The paper studies rigid sphere packings on 3D manifolds with boundary.

In this paper we develop an approach to conformal geometry of piecewise flat metrics on manifolds. In particular, we formulate the combinatorial Yamabe problem for piecewise flat metrics. In the case of surfaces, we define the combinatorial Yamabe flow on the space of all piecewise flat metrics associated to a triangul…

The paper applies combinatorial Ricci flows to prove hyperbolic structures on 3-manifolds.

Study combinatorial Yamabe flow on infinite triangulated surfaces.

We show existence of homothetically shrinking solutions of the fractional mean curvature flow, whose boundary consists in a prescribed numbers of concentric spheres. We prove that all these solutions, except from the ball, are dynamically unstable.

Study on deforming discrete conformal structures on surfaces with boundaries.

In this paper we consider the evolution of sets by a fractional mean curvature flow. Our main result states that for any dimension $n > 2$, there exists an embedded surface in $\mathbb R^n$ evolving by fractional mean curvature flow, which developes a singularity before it can shrink to a point. When $n > 3$ this resul…

We introduce a fractional Yamabe flow involving nonlocal conformally invariant operators on the conformal infinity of asymptotically hyperbolic manifolds, and show that on the conformal spheres $(\Sn, [g_{\Sn}])$, it converges to the standard sphere up to a Möbius diffeomorphism. This result allows us to obtain extinct…

We investigate the properties of the combinatorial Ricci flow for surfaces, both forward and backward -- existence, uniqueness and singularities formation. We show that the positive results that exist for the smooth Ricci flow also hold for the combinatorial one and that, moreover, the same results hold for a more gene…

This paper uses combinatorial Ricci flow to tackle Thurston's triangulation conjecture.

The study provides a criterion for fractional-linear integrals of geodesics on surfaces.