RCD(κ,2): Two-Dimensional Synthetic Curvature Spaces

• RCD(κ,2) spaces are defined by synthetic curvature-dimension conditions and infinitesimal Hilbertianity, ensuring rigorous analytic and geometric structure in 2D.
• They are characterized via optimal transport convexity and the Bakry–Émery formalism, allowing precise PDE analysis including weak unique continuation for harmonic functions.
• A sharp dichotomy between collapsed and non-collapsed cases highlights geometric rigidity in 2D, while exposing analytic challenges in higher dimensions.

RCD$(\kappa,2)$, denoting spaces satisfying the Riemannian Curvature-Dimension condition with lower Ricci curvature bound $\kappa$ and dimension parameter $2$, forms a foundational class in synthetic (metric-measure) geometry. These spaces are characterized both via optimal transport convexity (CD) plus infinitesimal Hilbertianity, and through the Bakry–Émery/Bochner formalism, and they exhibit a sharp dichotomy between “collapsed” and “non-collapsed” structures. The analysis of elliptic PDEs, especially unique continuation of harmonic functions, reveals significant distinctions between two-dimensional RCD spaces and their higher-dimensional counterparts.

1. Precise Characterizations of $\mathrm{RCD}(\kappa,2)$ Spaces

Let $(X,d,\mathfrak m)$ be a complete, separable metric space with a nonnegative Radon measure $\mathfrak m$ and $\operatorname{supp}\mathfrak m = X$. There are two equivalent formulations for the class $\mathrm{RCD}(\kappa,2)$:

(a) Curvature-Dimension plus Infinitesimal Hilbertianity

A space is $\mathrm{CD}(\kappa,2)$ if at every point $o\in X$, for some neighborhood $U\ni o$ and any pair of probability measures $\mu_0,\mu_1\ll\mathfrak m$ with supports in $U$, the $t$-interpolation $\mu_t = (\mathrm e_t)_\sharp\pi$ along an optimal coupling $\pi$ satisfies for all $N'\ge2$ and almost every $t\in[0,1]$ the inequality: $$\mathcal E_{N'}(\mu_t)\ge \int_{X\times X} \Big[\tau_{\kappa,2}^{(1-t)}(d(x_0,x_1))\,\rho_0(x_0)^{-1/N'} + \tau_{\kappa,2}^{(t)}(d(x_0,x_1))\,\rho_1(x_1)^{-1/N'}\Big]\,\mathrm d\pi(x_0,x_1),$$ where $\mu_i=\rho_i\mathfrak m$ and $$\mathcal E_{N'}(\mu)=\int\rho^{1-1/N'}\,\mathrm d\mathfrak m$$. The distortion coefficient $\tau_{\kappa,2}^{(t)}(\theta)$ takes the form: $\tau_{\kappa,2}^{(t)}(\theta) = \begin{cases} \frac{\sin(t\,\theta\sqrt{\kappa/2})}{\sin(\theta\sqrt{\kappa/2})} & \kappa>0, \[1ex] t & \kappa=0, \[1ex] \frac{\sinh(t\,\theta\sqrt{-\kappa/2})}{\sinh(\theta\sqrt{-\kappa/2})} & \kappa<0. \end{cases}$ Infinitesimal Hilbertianity, requiring $W^{1,2}(X)$ to be a Hilbert space, is enforced (e.g. via the Cheeger energy).

(b) Bakry–Émery/Bochner Formulation

Alternatively, $(X,d,\mathfrak m)$ is $\mathrm{RCD}(\kappa,2)$ if it is infinitesimally Hilbertian and its heat semigroup $(P_t)_{t\ge0}$ fulfills the weak Bochner inequality: for all $f\in D(\Delta)$ with $\Delta f\in W^{1,2}(X)$,

$$\frac{1}{2}\Delta|\nabla f|^2\geq \langle\nabla f,\nabla(\Delta f)\rangle + \kappa|\nabla f|^2 + \frac{1}{2}\frac{(\Delta f)^2}{2}$$

in the distributional sense. Here, $\Delta$ is the infinitesimal generator of the $L^2$ heat flow and $|\nabla f|$ is the minimal relaxed gradient.

2. Weak Unique Continuation for Harmonic Functions

A fundamental result for $\mathrm{RCD}(\kappa,2)$ spaces is the weak unique continuation theorem for harmonic functions. Precisely, if $(X,d,\mathfrak m)$ is an $\mathrm{RCD}(\kappa,2)$ space and $\Omega\subset X$ is connected, any $\varphi\in W^{1,2}(\Omega)$ solving in weak form the divergence-form elliptic equation

$$-\mathrm{div}(a\nabla\varphi) + \mathbf b\cdot\nabla\varphi + c\varphi = 0$$

with measurable locally bounded coefficients ($a\ge\gamma>0$), vanishing on a ball $B_r(x)$, must vanish identically on $\Omega$.

The proof relies on the classification of $\mathrm{RCD}(\kappa,2)$ spaces:

• In the non-collapsed case, $(X,d)$ is an Alexandrov surface with $\mathfrak m = \mathcal H^2$, admitting local $C^{1,\alpha}$ harmonic charts. Unique continuation then follows from the classical theory for elliptic equations with $L^\infty$ coefficients.
• In the collapsed case, the essential dimension is $1$ (the space is isometric to a line, ray, interval, or circle) and ODE theory applies.

The connectedness of vanishing sets and standard analytic arguments ensure global vanishing. No “fractal” singular behavior arises in dimension $2$.

3. Classification and Structure in Dimension Two

The dichotomy in the structure of $\mathrm{RCD}(\kappa,2)$ spaces is central:

Case	Description	PDE Implication
Non-collapsed	Alexandrov surface; $\mathfrak m = \mathcal H^2$; regular points admit harmonic charts	Classical strong & weak unique continuation
Collapsed	1-dimensional: isometric to line, ray, interval, or circle	ODE unique continuation

Regularity results—such as existence of $C^{1,\alpha}$ (in fact, $W^{2,p}$) harmonic coordinates—enable the transfer of classical PDE techniques. The absence of more exotic geometric singularities in dimension two justifies reducing PDE behavior to the well-understood Alexandrov or 1D cases.

A plausible implication is that, in two dimensions, Ricci curvature lower bounds in the synthetic sense are sufficient to enforce classical analytic properties of harmonic functions.

4. Breakdown of Strong Unique Continuation in Higher Dimensions

For $N\ge4$ and every $\kappa\in\mathbb R$, there exist $\mathrm{RCD}(\kappa,N)$ spaces $(X,d,\mathfrak m)$ and harmonic functions $u:\Omega\to\mathbb R$ on open sets $\Omega\subset X$ such that $u$ vanishes to infinite order at a point $x_0$ but is not identically zero—demonstrating the failure of strong unique continuation.

The construction proceeds via:

1. Warped-product metrics near a vertex: On $(0,r_*)\times S^{n-1}$, with $n=N-1\ge3$, the metric

$$g_{\varepsilon,n} = dr^2 + r^{2(1+\varepsilon)}g_{S^{n-1}}$$

and measure $$d\mathfrak m = r^{1-\eta} d\operatorname{vol}_{g_{\varepsilon,n}}$$ are selected to ensure $\operatorname{Ric}_N \ge \kappa$.

1. Gluing: For large $r$, the warping function is interpolated to secure a complete $\mathrm{CD}(\kappa,N)$ space, maintaining rotational symmetry and verifying infinitesimal Hilbertianity.
2. Nontrivial harmonic function construction: Via Perron’s method and a three-circle theorem (Colding–Minicozzi style), nontrivial harmonic functions vanishing at the tip are constructed and extracted as limits from tangent cone analysis at infinity.
3. Infinite order vanishing: The Cheng–Yau gradient estimate leads to

$$\sup_{B_r}|u|\leq C r^\alpha \sup_{B_{2r}}|u|,\quad\alpha>1$$

and, after iteration,

$|u(x)| = O(r^m),\ \forall m \in \mathbb N$

for $d(x,x_0) = r\ll 1$, with $u\not\equiv 0$.

This construction shows that singularities in higher-dimensional RCD spaces can defeat strong unique continuation, even as weak unique continuation may persist.

5. Analytic and Geometric Rigidity, Open Problems

Several consequences and open questions arise:

• Rigidity in dimension two: The synthetic Ricci condition enforces reduction to settings where classical unique continuation theorems hold; no pathological geometric phenomena are possible.
• Regularity of solutions: Regularity of the metric and control over analytic structure at regular points are sufficient for both weak and strong unique continuation in $2$D; frequency function methods can be implemented as in the smooth case.
• Singularities in higher dimensions: New singular phenomena in higher-dimensional spaces destroy strong unique continuation at singular points. Whether strong unique continuation persists at regular points in higher-dimensional settings remains open.
• Geometric thresholds and analytic prerequisites: Open questions include the identification of geometric or analytic thresholds (such as Hölder exponents for the metric) guaranteeing strong unique continuation, and potential consequences for geometric rigidity theorems (e.g., those in the Cheeger–Colding program) under extra analytic constraints.

In summary, for $\mathrm{RCD}(\kappa,2)$ spaces the synthetic Ricci curvature lower bound is sufficient to recover classical unique continuation theorems for harmonic functions, while higher-dimensional analogues display fundamental obstacles to strong unique continuation at singularities (Deng et al., 2022).