arXiv subjects

Aijin Lin

Publications and source records attributed to Aijin Lin.

13 recordsLinked to original sources

arXiv2026

Circle Pattern Theorem for Quasi-simplicial Triangulated Surfaces

The Circle Pattern Theorem characterizes the existence and rigidity of circle patterns with prescribed intersection angles on simplicial triangulations of closed surfaces. In this paper we extend the theorem to quasi-simplicial triangulations -- triangulations that may contain loops and multiple edges, but whose lifts to the universal cover are simplicial. Chow and Luo first considered such triangulations -- under the name \emph{generalized triangulations} (J.~Differential Geom.~\textbf{63}(1):97--129, 2003) -- but with the strong restriction that any three vertices determine at most one triangle; this condition keeps the combinatorics within the simplicial complex framework and consequently excludes most quasi-simplicial triangulations. We remove this restriction, work instead with the more flexible framework of Delta complexes, and use a finite covering technique to reduce the problem to the simplicial case. We prove that the curvature image is completely characterized by KAT inequalities imposed on all subsets of the lifted vertex set.

math.GT↗

arXiv2026

The character of ideal circle patterns

Let $S$ be an oriented closed surface with a cellular decomposition $\mathcal{D}$ and a weight $\Phi\in(0, \pi)$. It is crucial to determine when $S$ supports an ideal $\mathcal{D}$-type circle pattern $\mathcal{P}$ with the exterior intersection angles given by $\Phi$. Rivin, Bobenko-Springborn and Ge-Hua-Zhou provided perfect solutions and gave wonderful criteria for the existence and uniqueness of ideal circle patterns. However, all criteria established by Rivin, Bobenko-Springborn and Ge-Hua-Zhou are extremely difficult to verify for the given cellular decomposition $\mathcal{D}$ and the weight $\Phi$. In this paper, we introduce the character $\mathcal{L}(\mathcal{D},\Phi)$ depends only on the data of the weighted cellular decomposition $(\mathcal D, \Phi)$ on $S$, and give some quite simple criteria for the existence of ideal circle patterns realizing $(\mathcal{D},\Phi)$. It seems that our character-type criteria are the first conditions totally different from criteria of Rivin, Bobenko-Springborn and Ge-Hua-Zhou, and provide more easily verifiable criteria. Our new character-type theorems may be of some independent interest. As an application, we give a new descriptions of the curvature image set $\mathbf{K}(\mathbb{R}^N_{>0})$. To approach our results, we shall use the combinatorial Ricci flows with ideal circle patterns introduced by Ge-Hua-Zhou as a fundamental tool. The main difficulty in the proof of our results is to establish the compactness of the solution to the flows. To circumvent the difficulty, we borrow the techniques developed by Ge and his collaborators.

math.DG↗

arXiv2025

Improved explicit estimates for the discrete Laplace operator with hyperbolic circle patterns

Ge in his thesis \cite{Ge-thesis} introduced the combinatorial Calabi flows and established the long time existence and convergence of solutions to the flows in both hyperbolic and Euclidean background geometries. It is noteworthy that the existence of solutions to the combinatorial Calabi flows in hyperbolic background geometry proves to be more intricate and challenging compared to the Euclidean background geometry. The main difficulty is to establish the compactness, especially the lower boundeness along the flow equations. In this paper, we give two explicit estimates for the discrete Laplace operator based on the Glickenstein-Thomas formulation \cite{Glickenstein2017} for discrete hyperbolic conformal structures. As applications, we give new proofs of the long time existence of solutions to the combinatorial Calabi flows established by Ge-Xu \cite{Ge2016}, Ge-Hua \cite{Ge2018} and the combinatorial $p$-th Calabi flows established by Lin-Zhang \cite{Lin2019} in hyperbolic background geometry.

math.DG↗

arXiv2025

Branched $\alpha$-combinatorial Ricci flows on closed surfaces with Euler characteristic $\chi\le 0$

In this paper we introduce the branched $\alpha$-flows on closed surfaces with Euler characteristic $\chi \leq 0$. Based on the strict convexity of the branched $\alpha$-potentials, we establish the long time existence and convergence of the solutions to the branched $\alpha$-flows, which generalizes Ge and Xu's main results \cite{2015,2015A} on the $\alpha$-flows. In addtion, we study the prescribed curvature problems under the relaxed precondition $\chi(M)\in \mathbb{Z}$ via alternative $\alpha$-flows, establishing admissibility conditions for prescribed curvatures and their exponential convergence to target metrics.

math.DG↗

arXiv2018

Combinatorial $p$-th Calabi flows on surfaces

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 different approaches, we show that the solution to the combinatorial $p$-th Calabi flow exists for all time and converges if and only if there exists a circle packing metric of constant (zero resp.) curvature in Euclidean (hyperbolic resp.) background geometry. Our results generalize the work of H. Ge, Ge-Xu and Ge-Hua on the combinatorial Calabi flow from $p=2$ to any $p>1$.

math.DG↗

arXiv2018

The K\"ahler-Ricci flow on pseudoconvex domains

We establish the existence of K\"ahler-Ricci flow on pseudoconvex domains with general initial metric without curvature bounds. Moreover we prove that this flow is simultaneously complete, and its normalized version converge to the complete K\"ahler-Einstein metric, which generalizes Topping's works on surfaces.

math.DG↗

arXiv2018

Gradient flow of the norm squared of a moment map over Kahler manifolds

Inspired by Wilkin's work [23, 24] on Morse theory for the moduli space of Higgs bundles, we study the moduli space of gauged holomorphic maps by a heat flow approach in the spirit of Atiyah and Bott in a series of papers. In this paper, applying the method of Hong [9], we establish the global existence of smooth solutions of the gradient ow equations of the vortex functional over a compact Kahler manifold.

math.DG↗

arXiv2017

Positive Solutions of p-th Yamabe Type Equations on Infinite Graphs

Let $G=(V,E)$ be a connected infinite and locally finite weighted graph, $\Delta_p$ be the $p$-th discrete graph Laplacian. In this paper, we consider the $p$-th Yamabe type equation $$-\Delta_pu+h|u|^{p-2}u=gu^{\alpha-1}$$ on $G$, where $h$ and $g$ are known, $2<\alpha\leq p$. The prototype of this equation comes from the smooth Yamabe equation on an open manifold. We prove that the above equation has at least one positive solution on $G$.

math.AP↗

arXiv2017

Mean Curvature Type Flows of Graphs in Product Manifolds

In this note we study a large class of mean curvature type flows of graphs in product manifold $N\times R$ where N is a closed Riemann- ian manifold. Their speeds are the mean curvature of graphs plus a prescribed function. We establish long time existence and uniformly convergence of those flows with a barrier condition and a condition on the derivative of prescribed function with respect to the height. As an application we construct a weighted mean curvature flow in large classes of warped product manifolds which evolves each graph into a totally ge- odesic slice

math.DG↗

arXiv2017

Positive Solutions of $p$-th Yamabe Type Equations on Graphs

Let $G=(V,E)$ be a finite connected weighted graph, and assume $1\leq\alpha\leq p\leq q$. In this paper, we consider the following $p$-th Yamabe type equation $$-\Delta_pu+hu^{q-1}=\lambda fu^{\alpha-1}.$$ on $G$, where $\Delta_p$ is the $p$-th discrete graph Laplacian, $h\leq0$ and $f>0$ are real functions defined on all vertices of $G$. Instead of the approach in [Ge3], we adopt a new approach, and prove that the above equation always has a positive solution $u>0$ for some constant $\lambda\in\mathbb{R}$. In particular, when $q=p$ our result generalizes the main theorem in [Ge3] from the case of $\alpha\geq p>1$ to the case of $1\leq\alpha\leq p$. It's interesting that our new approach can also work in the case of $\alpha\geq p>1$.

math.DG↗

arXiv2016

Conic K\"{a}hler-Einstein metrics along simple normal crossing divisors on Fano manifolds

We prove that on one K\"{a}hler-Einstein Fano manifold without holomorphic vector fields, there exists a unique conical K\"{a}hler-Einstein metric along a simple normal crossing divisor with admissible prescribed cone angles. We also establish a curvature estimate for conic metrics along a simple normal crossing divisor which generalizes Li-Rubinstein's estimate and derive high order estimates from this estimate.

math.DG↗

arXiv2012

The heat flow for Kahler fibrations

We establish global existence of smooth solutions to heat flow for Yang-Mills-Higgs functional on Kahler fibrations. As an application, we give a new proof of the key inequality for Mundet's Hitchin-Kobayashi correspondence theorem using the heat flow technique.

math.DG↗

arXiv2012

Index computations in FJRW theory

We compute the index of the real Cauchy-Riemann operator defined in FJRW theory in case of the smooth metric. For the cylindrical metric, we study the relation between the index of the linearized operator of Witten map and weights in weighted Sobolev space.

math.DG↗