Discrete Yamabe Problem

• Discrete Yamabe Problem is the discrete analogue of the classical Yamabe problem that seeks constant curvature metrics through combinatorial and variational methods.
• It reformulates smooth concepts like curvature, Laplacians, and conformal factors into discrete counterparts on polyhedral surfaces and graphs.
• The approach employs variational techniques and graph-based elliptic equations, highlighting issues of existence, uniqueness, and compactness in non-smooth settings.

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 (Chen, 2023). 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 (Kouřimská, 2021).

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 (Brendle et al., 2010). 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 (Chen, 2023). 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 (Brendle et al., 2010).

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 (Brendle et al., 2010). 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 (Brendle et al., 2010).

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” (Brendle et al., 2010). 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 (Kouřimská, 2021). The motivating difficulty is that the standard discrete curvature quantity in geometry processing, the angle defect

$(M,g_0)$0

is intrinsic but does not scale like smooth Gaussian curvature under global rescaling. Because smooth Gaussian curvature scales as $(M,g_0)$1, the polyhedral theory introduces a curvature density rather than using the defect alone.

For a conical singularity $(M,g_0)$2, let $(M,g_0)$3 be the area of the Voronoi cell of $(M,g_0)$4. The discrete Gaussian curvature is then defined by

$(M,g_0)$5

This definition is the key new ingredient of the polyhedral theory (Kouřimská, 2021). It is intrinsic, satisfies a Gauss–Bonnet formula, and under global scaling by $(M,g_0)$6 transforms as $(M,g_0)$7. 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 (Kouřimská, 2021).

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 (Kouřimská, 2021). On a fixed triangulation $(M,g_0)$8, two discrete metrics $(M,g_0)$9 are discretely conformally equivalent if there exists $n\ge 3$0 such that

$n\ge 3$1

This is the direct discrete analogue of the smooth conformal law $n\ge 3$2.

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 (Kouřimská, 2021). Two PL metrics $n\ge 3$3 on $n\ge 3$4 are discretely conformally equivalent if the decorated hyperbolic surfaces they induce are isometric by a map homotopic to the identity on $n\ge 3$5 relative to $n\ge 3$6. The conformal class is parametrized by

$n\ge 3$7

and for each $n\ge 3$8 one obtains a new metric $n\ge 3$9. The subset on which a triangulation $g_0$0 remains Delaunay is the Penner cell

$g_0$1

Within this framework, the central theorem is the discrete uniformization theorem: for every PL metric $g_0$2 on a marked surface $g_0$3, there exists a discrete conformally equivalent PL metric $g_0$4 such that $g_0$5 has constant discrete Gaussian curvature (Kouřimská, 2021). Every discrete conformal class therefore contains at least one representative for which the values

$g_0$6

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: $g_0$7, whose partial derivatives satisfy

$g_0$8

and the total area $g_0$9, whose partial derivatives satisfy

$S^n$0

(Kouřimská, 2021). By Lagrange multipliers, critical points of $S^n$1 under the constraint $S^n$2 correspond exactly to metrics with constant

$S^n$3

Equivalently, one can work with

$S^n$4

whose critical points are precisely the constant-discrete-curvature metrics.

The existence argument is sensitive to the sign of $S^n$5. In the spherical case $S^n$6 and hyperbolic case $S^n$7, $S^n$8 attains a minimum on the appropriate constraint set; the Euclidean case $S^n$9 is handled by previously known discrete uniformization results (Kouřimská, 2021). 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 (Kouřimská, 2021). Uniqueness is proved only in three special cases: genus $\partial_t g(t)=-(R_{g(t)}-\rho(t))\,g(t),$0 with $\partial_t g(t)=-(R_{g(t)}-\rho(t))\,g(t),$1, genus $\partial_t g(t)=-(R_{g(t)}-\rho(t))\,g(t),$2, and genus $\partial_t g(t)=-(R_{g(t)}-\rho(t))\,g(t),$3 with $\partial_t g(t)=-(R_{g(t)}-\rho(t))\,g(t),$4. 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

$\partial_t g(t)=-(R_{g(t)}-\rho(t))\,g(t),$5

produces multiple constant-curvature metrics for distinct values of $\partial_t g(t)=-(R_{g(t)}-\rho(t))\,g(t),$6. A genus-two surface with two marked points yields an analogous phenomenon through

$\partial_t g(t)=-(R_{g(t)}-\rho(t))\,g(t),$7

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 (Grigor'yan et al., 2016, Ge, 2016).

On a locally finite graph $\partial_t g(t)=-(R_{g(t)}-\rho(t))\,g(t),$8 with positive symmetric edge weights $\partial_t g(t)=-(R_{g(t)}-\rho(t))\,g(t),$9 and positive vertex measure $(S,V,d)$0, the graph Laplacian is defined by

$(S,V,d)$1

(Grigor'yan et al., 2016). For a bounded connected domain $(S,V,d)$2, with graph-theoretic boundary $(S,V,d)$3 and interior $(S,V,d)$4, the main semilinear equation is

$(S,V,d)$5

If $(S,V,d)$6 and $(S,V,d)$7, there exists a positive solution (Grigor'yan et al., 2016). The proof uses the mountain pass theorem of Ambrosetti–Rabinowitz applied to

$(S,V,d)$8

On finite connected graphs, the $(S,V,d)$9-Laplacian version takes the form

$S$0

where

$S$1

$S$2, and $S$3 (Ge, 2016). In this setting there always exists a positive solution $S$4 for some constant $S$5. The proof is by direct minimization of

$S$6

Setting	Core equation	Existence statement
Bounded domain in a locally finite graph	$S$7, $S$8 on $S$9	Positive solution if $V\subset S$0 and $V\subset S$1 (Grigor'yan et al., 2016)
Finite connected graph	$V\subset S$2	Positive solution for some $V\subset S$3 if $V\subset S$4 and $V\subset S$5 (Ge, 2016)

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 (Grigor'yan et al., 2016). They differ, however, from the polyhedral-surface problem: the central object is a discrete elliptic equation rather than a curvature density $V\subset S$6, 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 $V\subset S$7-mass at points [(Brendle et al., 2010); (Chen, 2023)]. 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 (Grigor'yan et al., 2016, Ge, 2016). In the polyhedral-surface theory discussed above, the prominent subtlety is not blow-up but non-uniqueness within a discrete conformal class (Kouřimská, 2021).

A broader analytic generalization appears on Dirichlet spaces. There the Yamabe problem is formulated through a closed symmetric bilinear form $V\subset S$8, a Schrödinger-type operator $V\subset S$9, and the critical invariant

$d$0

with solvability under the strict inequality

$d$1

where $d$2 is a Sobolev constant (Akutagawa et al., 2013). 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 (Xu, 2024). 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.