---
title: Discrete Yamabe Problem
url: https://www.emergentmind.com/topics/discrete-yamabe-problem
type: topic
---

# Discrete Yamabe Problem

The discrete Yamabe problem is the discrete analogue of the classical Yamabe problem, in which one seeks a conformal deformation to constant curvature. In the smooth setting, the problem asks whether a compact Riemannian manifold admits a metric in a given conformal class with constant scalar curvature; under a conformal change \(\tilde g=\phi^{4/(n-2)}g\), the variational formulation leads to an Euler–Lagrange equation of Yamabe type [2309.02397]. In discrete settings, the same program is reformulated on piecewise-flat or combinatorial structures by replacing smooth metrics, curvature densities, Laplace operators, and conformal factors with discrete counterparts. In the literature represented here, the term encompasses both an exact two-dimensional analogue on polyhedral surfaces and graph-based Yamabe-type equations that retain the nonlinear variational structure of the smooth theory [2103.15693].

## 1. Continuous model and conceptual template

The smooth Yamabe problem is framed as the higher-dimensional analogue of uniformization: given a compact Riemannian manifold \((M,g_0)\) of dimension \(n\ge 3\), one seeks a metric in the conformal class of \(g_0\) with constant scalar curvature [1010.4960]. In variational terms, the problem is governed by the Yamabe functional, and a minimizer yields a conformal metric of constant scalar curvature through the associated Euler–Lagrange equation [2309.02397]. The sphere \(S^n\) furnishes the sharp comparison value, and the strict inequality against the spherical constant is the classical mechanism that rules out concentration and produces existence [1010.4960].

Two structural features of the continuous theory are especially important. First, compactness can fail through bubbling: concentrating sequences resemble rescaled standard spheres, and the compactness theory becomes dimension-dependent [1010.4960]. Second, the problem admits both elliptic and parabolic formulations, the latter through the Yamabe flow
\[
\partial_t g(t)=-(R_{g(t)}-\rho(t))\,g(t),
\]
which evolves metrics toward constant scalar curvature in several settings [1010.4960].

The survey literature makes the discrete implication explicit: for the discrete Yamabe problem, the continuous theory suggests the right conceptual template is “conformal change of metric, scalar-curvature-preserving equations, variational structure, bubbling/compactness issues, and flow-based regularization” [1010.4960]. This suggests that any discrete formulation must specify, at minimum, a notion of discrete conformal deformation, a discrete curvature density rather than a purely integrated defect, and an energy whose critical points represent constant-curvature configurations.

## 2. Polyhedral surfaces and discrete curvature density

A direct formulation of the discrete Yamabe problem is given for polyhedral surfaces \((S,V,d)\), where \(S\) is a closed oriented surface, \(V\subset S\) is a finite set of marked points containing the conical singularities, and \(d\) is a piecewise flat metric [2103.15693]. The motivating difficulty is that the standard discrete curvature quantity in geometry processing, the angle defect
\[
W_i=2\pi-\alpha_i,
\]
is intrinsic but does not scale like smooth Gaussian curvature under global rescaling. Because smooth Gaussian curvature scales as \(r^{-2}\), the polyhedral theory introduces a curvature density rather than using the defect alone.

For a conical singularity \(i\in V\), let \(A_i\) be the area of the Voronoi cell of \(i\). The discrete Gaussian curvature is then defined by
\[
K_i=\frac{W_i}{A_i}=\frac{2\pi-\alpha_i}{A_i}.
\]
This definition is the key new ingredient of the polyhedral theory [2103.15693]. It is intrinsic, satisfies a Gauss–Bonnet formula, and under global scaling by \(r\) transforms as \(K_i\mapsto r^{-2}K_i\). In this way, the angle defect is converted into a curvature density, paralleling the smooth relation between total curvature and pointwise Gaussian curvature.

The discrete Yamabe problem in this setting asks: given a polyhedral surface, does there exist a discretely conformally equivalent polyhedral metric with constant discrete Gaussian curvature? The question is a direct analogue of two-dimensional uniformization, where every closed oriented smooth surface is conformally equivalent to one with constant Gaussian curvature [2103.15693].

## 3. Discrete conformal classes and the uniformization theorem

The polyhedral theory uses a generalization of discrete conformal equivalence pioneered by Feng Luo and developed further by Bobenko–Pinkall–Springborn and others [2103.15693]. On a fixed triangulation \(\Delta\), two discrete metrics \(\ell,\tilde\ell:E_\Delta\to \mathbb{R}_{>0}\) are discretely conformally equivalent if there exists \(u:V\to\mathbb{R}\) such that
\[
\tilde{\ell}_{ij}=\exp\!\left(\frac{u_i+u_j}{2}\right)\ell_{ij}
\qquad \text{for every edge }ij\in E_\Delta.
\]
This is the direct discrete analogue of the smooth conformal law \(\tilde g=e^{2u}g\).

The formulation used for polyhedral surfaces is more general than a fixed triangulation. It passes through decorated hyperbolic surfaces with cusps: piecewise flat metrics correspond to Penner coordinates, and Delaunay triangulations correspond to ideal Delaunay triangulations [2103.15693]. Two PL metrics \(d,\tilde d\) on \((S,V)\) are discretely conformally equivalent if the decorated hyperbolic surfaces they induce are isometric by a map homotopic to the identity on \(S\setminus V\) relative to \(V\). The conformal class is parametrized by
\[
R^V=\{u:V\to\mathbb{R}\},
\]
and for each \(u\in R^V\) one obtains a new metric \(d(u)\). The subset on which a triangulation \(\Delta\) remains Delaunay is the Penner cell
\[
\mathcal{A}_\Delta=\{u\in R^V\mid \Delta \text{ is a Delaunay triangulation of }(S,V,d(u))\}.
\]

Within this framework, the central theorem is the discrete uniformization theorem: for every PL metric \(d\) on a marked surface \((S,V)\), there exists a discrete conformally equivalent PL metric \(\tilde d\) such that \((S,V,\tilde d)\) has constant discrete Gaussian curvature [2103.15693]. Every discrete conformal class therefore contains at least one representative for which the values
\[
K_i=\frac{2\pi-\alpha_i}{A_i}
\]
are the same at all marked points. This is the exact two-dimensional discrete analogue of uniformization/Yamabe existence.

## 4. Variational structure and the failure of uniqueness

The existence proof for polyhedral surfaces is variational. Two functionals control the geometry: \(\mathbb{E}(u)\), whose partial derivatives satisfy
\[
\frac{\partial \mathbb{E}}{\partial u_i}=W_i,
\]
and the total area \(A_{\mathrm{tot}}(u)\), whose partial derivatives satisfy
\[
\frac{\partial A_{\mathrm{tot}}}{\partial u_i}=2A_i
\]
[2103.15693]. By Lagrange multipliers, critical points of \(\mathbb{E}\) under the constraint \(A_{\mathrm{tot}}(u)=1\) correspond exactly to metrics with constant
\[
\frac{W_i}{A_i}=K_i.
\]
Equivalently, one can work with
\[
F(u)=\mathbb{E}(u)-\pi\chi(S)\log(A_{\mathrm{tot}}(u)),
\]
whose critical points are precisely the constant-discrete-curvature metrics.

The existence argument is sensitive to the sign of \(\chi(S)\). In the spherical case \(\chi(S)=2\) and hyperbolic case \(\chi(S)<0\), \(\mathbb{E}\) attains a minimum on the appropriate constraint set; the Euclidean case \(\chi(S)=0\) is handled by previously known discrete uniformization results [2103.15693]. The overall structure is therefore closely parallel to the smooth Yamabe strategy: define a conformally natural energy, normalize scale, and identify constant curvature as the critical-point condition.

A major difference from smooth normalized uniformization is that the discrete constant-curvature representative is not unique in general [2103.15693]. Uniqueness is proved only in three special cases: genus \(0\) with \(|V|=3\), genus \(1\), and genus \(>1\) with \(|V|=1\). Outside these regimes, explicit counterexamples show that one discrete conformal class can contain more than one constant-curvature PL metric. On a tetrahedral sphere with four marked points, the family
\[
u(v)=(0,0,v,v)
\]
produces multiple constant-curvature metrics for distinct values of \(v\). A genus-two surface with two marked points yields an analogous phenomenon through
\[
u(v)=(0,v).
\]
A common misconception is therefore that discrete uniformization always singles out a canonical representative. In the polyhedral setting considered here, existence is universal but canonicity generally fails.

## 5. Graph-based Yamabe-type equations

A second line of work treats the discrete Yamabe problem through nonlinear elliptic equations on graphs. Here the emphasis is not an exact two-dimensional uniformization statement but a class of Yamabe-type equations that preserve the variational and positivity structure of the smooth theory [1607.04521; 1611.04906].

On a locally finite graph \(G=(V,E)\) with positive symmetric edge weights \(W_{xy}\) and positive vertex measure \(\mu\), the graph Laplacian is defined by
\[
\Delta u(x)=\frac{1}{\mu(x)}\sum_{y\sim x}W_{xy}\big(u(y)-u(x)\big)
\]
[1607.04521]. For a bounded connected domain \(\Omega\subset V\), with graph-theoretic boundary \(\partial\Omega\) and interior \(\Omega^\circ\), the main semilinear equation is
\[
-\Delta u-\alpha u=|u|^{p-2}u
\quad \text{in }\Omega^\circ,
\qquad
u=0 \text{ on }\partial\Omega.
\]
If \(\alpha<\lambda_1(\Omega)\) and \(p>2\), there exists a positive solution [1607.04521]. The proof uses the mountain pass theorem of Ambrosetti–Rabinowitz applied to
\[
J(u)=\frac12\int_\Omega \left(|\nabla u|^2-\alpha u^2\right)\,d\mu-\frac1p\int_\Omega (u^+)^p\,d\mu.
\]

On finite connected graphs, the \(p\)-Laplacian version takes the form
\[
\Delta_p \varphi + h\,\varphi^{p-1}=\lambda f\,\varphi^{\alpha-1},
\]
where
\[
\Delta_p u(i)=\sum_{j\sim i} w_{ij}\,|u_j-u_i|^{p-2}(u_j-u_i),
\]
\(f>0\), and \(\alpha>p>1\) [1611.04906]. In this setting there always exists a positive solution \(\varphi\) for some constant \(\lambda\in\mathbb{R}\). The proof is by direct minimization of
\[
I(\varphi)=
\frac{\displaystyle \int_E |\nabla \varphi|^p\,dw+\int_V h\varphi^p\,d\mu}
{\left(\displaystyle \int_V f\varphi^\alpha\,d\mu\right)^{p/\alpha}}.
\]

| Setting | Core equation | Existence statement |
|---|---|---|
| Bounded domain in a locally finite graph | \(-\Delta u-\alpha u=|u|^{p-2}u\), \(u=0\) on \(\partial\Omega\) | Positive solution if \(\alpha<\lambda_1(\Omega)\) and \(p>2\) [1607.04521] |
| Finite connected graph | \(\Delta_p \varphi+h\varphi^{p-1}=\lambda f\varphi^{\alpha-1}\) | Positive solution for some \(\lambda\) if \(\alpha>p>1\) and \(f>0\) [1611.04906] |

These graph problems are discrete analogues in the sense that the Laplace operator is replaced by a graph Laplacian, the geometry is encoded combinatorially, and the same superlinear nonlinearities and constrained variational principles appear [1607.04521]. They differ, however, from the polyhedral-surface problem: the central object is a discrete elliptic equation rather than a curvature density \(K_i=W_i/A_i\), and the main theorem is an existence statement for positive solutions rather than a discrete uniformization theorem.

## 6. Compactness, thresholds, and broader non-smooth formulations

Compactness occupies a different role across the various discrete models. In the smooth Yamabe problem, compactness can fail by bubbling, and the sphere provides the extremal threshold; concentration-compactness identifies the loss of compactness as concentration of critical \(L^{2^*}\)-mass at points [1010.4960; 2309.02397]. By contrast, on bounded graph domains and finite graphs, the relevant Sobolev spaces are finite-dimensional or pre-compact, so the existence arguments do not require concentration-compactness [1607.04521; 1611.04906]. In the polyhedral-surface theory discussed above, the prominent subtlety is not blow-up but non-uniqueness within a discrete conformal class [2103.15693].

A broader analytic generalization appears on Dirichlet spaces. There the Yamabe problem is formulated through a closed symmetric bilinear form \(\mathcal E\), a Schrödinger-type operator \(L+V\), and the critical invariant
\[
Y(V)=\inf\left\{\mathcal E_V(u): u\in D(\mathcal E),\ \|u\|_{L^{\frac{2\nu}{\nu-2}}}=1\right\},
\]
with solvability under the strict inequality
\[
Y(V)<\frac{1}{A},
\]
where \(A\) is a Sobolev constant [1306.4373]. The decisive mechanism is again a gap between a global invariant and a local obstruction. This suggests that the most robust abstract form of the discrete Yamabe problem is not tied to a single combinatorial model, but to a triad consisting of a discrete energy, a critical normalization, and a local threshold that prevents concentration or degeneration.

Recent smooth work also emphasizes a local variational method based on compactly supported test functions and comparison with the best Sobolev constant, rather than Green’s functions or the positive mass theorem [2410.13537]. That paper explicitly notes that this perspective is especially suggestive for a discrete Yamabe problem because it replaces global analytic inputs by local test-function estimates and comparison with a model constant. Taken together with the continuous survey literature, this indicates that discrete Yamabe theory is best understood as a family of conformally natural critical problems whose main invariants are existence, compactness, uniqueness, and threshold behavior, realized differently on polyhedral surfaces, graphs, and more general non-smooth spaces.

Source: https://www.emergentmind.com/topics/discrete-yamabe-problem