(Completed) Prescribed Gaussian Curvature by Discrete Conformality

Project Overview

This project gives a discrete conformal geometry approach to approximate solutions of the classical prescribed Gaussian curvature problem on closed, orientable smooth surfaces of genus > 1 (the negatively curved regime). The method turns the smooth conformal PDE into a finite-dimensional convex variational problem on a geodesic triangulation. The target discrete solution is a vertex conformal factor $u \in \mathbb{R}^V$ that can be computed efficiently by minimizing a globally convex functional, and it converges to the smooth conformal factor with an explicit first-order error bound.

Status: Completed (paper-level result).

Problem Formulation (Smooth Geometry)

Let $(S, g)$ be a connected closed orientable smooth Riemannian surface with genus $> 1$. Given a smooth target curvature function

the prescribed Gaussian curvature problem asks for a metric $\tilde{g}$ conformal to $g$ such that the Gaussian curvature of $(S, \tilde{g})$ equals $\tilde{\kappa}$.

Writing the conformal change as

the unknown conformal factor $\tilde{u}$ solves the curvature equation

where $\kappa$ is the Gaussian curvature of $(S,g)$ and $\Delta_g$ is the Laplace–Beltrami operator.

In the regime $\text{genus}(S) > 1$ and $\tilde{\kappa} < 0$, the smooth problem has a unique smooth solution $\tilde{u}$ (classical results of Berger / Kazdan–Warner).

Discrete Model: Geodesic Triangulation + Vertex Scaling

1) Geodesic triangulation

Let $T$ be a geodesic triangulation of $(S,g)$, with vertex/edge/face sets:

• $V = V(T)$
• $E = E(T)$
• $F = F(T)$

Let the (geodesic) edge lengths measured in $(S,g)$ be

We regard $(T, l)$ as a polyhedral approximation of $(S,g)$.

2) Discrete conformal factor (vertex scaling)

A discrete conformal factor is a vector

It rescales edge lengths by the standard vertex scaling rule:

This is the discrete conformality notion developed (in different curvature backgrounds) by Luo and by Bobenko–Pinkall–Springborn.

3) “Mixed” negative curvature background per face

To encode the target curvature field $\tilde{\kappa}$, assign to each triangle $\sigma \in F$ a constant negative curvature

chosen by sampling $\tilde{\kappa}$ on that face, e.g.

Each face is then realized as a geodesic triangle in constant curvature $K(\sigma)$ with side lengths given by the scaled edges $(u * l)$.

Discrete Curvature and the Target Discrete Equation

For each triangle $\triangle ijk$ with face curvature $K(\triangle ijk)$, compute its inner angles $\theta^i_{jk}(u)$ at vertex $i$ (these depend on $K(\triangle ijk)$ and the scaled edge lengths).

Define the generalized discrete curvature at each vertex $i \in V$ by angle defect:

The discrete prescribed curvature problem in this project is:

Find $u \in \mathbb{R}^V$ such that

\[K(u) = 0 \quad \text{(i.e., } K_i(u)=0 \text{ for all } i \in V).\]

Interpretation: the face-wise constants $K(\sigma)$ encode the desired curvature field, and the condition $K(u)=0$ enforces global compatibility of the glued metric so that the resulting polyhedral surface approximates the smooth solution metric $e^{2\tilde{u}} g$.

What We Want (Deliverables)

1. Discrete well-posedness: show $K(u)=0$ has a unique solution under standard mesh regularity and refinement.
2. Efficient computation: show the solution can be obtained by minimizing a globally convex functional (no spurious local minima).
3. Convergence with rate: prove $u$ converges to the smooth solution $\tilde{u}$ restricted to vertices, with an explicit first-order error bound.

Method: Convex Variational Principle + Discrete Elliptic Control

A) Convex energy whose gradient is curvature

A key structural fact is that the 1-form

is closed on the domain where the discrete triangles are well-defined, so one can define a potential

which is locally strictly convex.

Moreover, Bobenko–Pinkall–Springborn provide an explicit extension to a globally convex functional $\tilde{F}$ defined on all of $\mathbb{R}^V$. Consequently:

• $K(u)=0$ has at most one solution.
• The solution can be computed efficiently by minimizing $\tilde{F}(u)$.

B) The “Jacobian” structure: diagonal minus weighted graph Laplacian

A central technical tool is an explicit formula for the differential of discrete curvature:

where

• $D(u)$ is a positive diagonal matrix,
• $\Delta_{\eta(u)}$ is a weighted graph Laplacian on $(V,E)$ with edge weights $\eta_{ij}(u)$.

This is the discrete analog of ellipticity (and it is exactly what enables stable Newton-type solvers and max-norm estimates).

C) Discrete elliptic estimate on graphs (Wu–Zhu)

The convergence proof relies on a sharp discrete elliptic estimate under a uniform C-isoperimetric condition on the triangulation graph. This estimate controls solutions to linear systems of the form

D) A continuation path to reach K(u)=0

The proof constructs a smooth path $u(t)$ on $[0,1]$ starting from the smooth solution restricted to vertices:

• $u(0) = \tilde{u}

  _{V}$,

such that

• $K(u(1)) = 0$,

• $u(1) - \tilde{u}

  _V = \mathcal{O}(

  l

  )$,

so $u(1)$ is the desired discrete conformal factor.

Main Theorem (Existence, Uniqueness, and Convergence Rate)

A polyhedral surface $(T,l)$ is called $\epsilon$-acute if every triangle angle satisfies

Theorem (Main Result). Let $(S,g)$ be closed orientable smooth with genus $> 1$. Let $\tilde{\kappa}(x) < 0$ be smooth on $S$, and let $\tilde{u}$ solve the smooth curvature equation so that $e^{2\tilde{u}} g$ has curvature $\tilde{\kappa}$. Let $T$ be a geodesic triangulation with edge lengths $l$, and set for each face $\sigma$ a constant $K(\sigma) = \tilde{\kappa}(x_\sigma)$ for some $x_\sigma \in \sigma$.

1. (Existence & uniqueness) there exists a unique $u \in \mathbb{R}^V$ such that \(K(K, u * l) = 0,\)
2. (First-order convergence) \(\bigl| u - \tilde{u}|_{V} \bigr| \le C\, |l|.\)

Practical Computation (Solver-Friendly Summary)

Inputs

• A geodesic triangulation $T$ of $(S,g)$ with edge lengths $l$.
• A smooth negative target curvature $\tilde{\kappa}$ sampled per face as $K(\sigma)$.

Unknown

• Vertex vector $u \in \mathbb{R}^V$.

Core loop (conceptual)

1. Compute scaled lengths $l’ = u * l$.
2. For each triangle $\sigma = ijk$, compute angles using the constant-curvature triangle geometry with curvature $K(\sigma)$ and side lengths $l’$.
3. Assemble vertex curvatures $K_i(u) = 2\pi - \sum \text{angles}$.
4. Solve $K(u)=0$ either by:
   • minimizing the globally convex energy $\tilde{F}(u)$, or
   • Newton updates using $\partial K/\partial u = D - \Delta_\eta$ (with damping/line-search if desired).

Because the objective is globally convex, the optimization landscape is well-behaved.

Selected References (Accurate Titles)

1. Melvyn S. Berger. Riemannian structures of prescribed gaussian curvature for compact 2-manifolds. Journal of Differential Geometry, 5(3–4):325–332, 1971.
2. Alexander I. Bobenko, Ulrich Pinkall, Boris A. Springborn. Discrete conformal maps and ideal hyperbolic polyhedra. Geometry & Topology, 19(4):2155–2215, 2015.
3. David Gu, Feng Luo, Tianqi Wu. Convergence of discrete conformal geometry and computation of uniformization maps. Asian Journal of Mathematics, 23(1):21–34, 2019.
4. Jerry L. Kazdan, Frank W. Warner. Curvature functions for compact 2-manifolds. Annals of Mathematics, 99(1):14–47, 1974.
5. Jerry L. Kazdan, Frank W. Warner. Scalar curvature and conformal deformation of riemannian structure. Journal of Differential Geometry, 10(1):113–134, 1975.
6. Feng Luo. Combinatorial yamabe flow on surfaces. Communications in Contemporary Mathematics, 6(05):765–780, 2004.
7. Tianqi Wu, Xiaoping Zhu. The convergence of discrete uniformizations for closed surfaces. Journal of Differential Geometry, 127(3):1305–1343, 2024.