Curvature-Weighted Poisson Equation

• The curvature-weighted Poisson equation is an inhomogeneous elliptic PDE framework that uses functionally adapted weights reflecting curvature decay on non-compact Riemannian manifolds.
• Weighted Poincaré inequalities decouple solution existence from global spectral or curvature bounds by quantifying local geometric degeneration.
• The approach unifies classical results across Euclidean, hyperbolic, and Cartan–Hadamard geometries, providing sharp solvability and decay estimates.

The curvature-weighted Poisson equation refers to a class of inhomogeneous elliptic partial differential equations on non-compact Riemannian manifolds, in which solvability and quantitative estimates are governed by weighted Poincaré inequalities incorporating geometric features such as curvature at infinity. The theory decouples the existence and uniqueness of solutions from classical curvature or spectral gap requirements, instead leveraging functionally adapted weights that reflect the degeneration or growth of the manifold's Ricci curvature and the asymptotic decay of the source term. This approach produces sharp solvability criteria and quantitative a priori estimates that recover and generalize traditional results for Euclidean, hyperbolic, and Cartan–Hadamard geometries (Catino et al., 2019, Munteanu et al., 2019).

1. Weighted Poincaré Inequalities and Definitions

Let $(M^n,g)$ denote a complete non-compact Riemannian manifold with Laplace–Beltrami operator $\Delta_g$. The inhomogeneous Poisson equation is

$\Delta_g u = f \quad \text{on } M,$

where $f$ is the prescribed source. Solvability on non-compact $M$ generally fails unless one imposes spectral gap conditions ($\lambda_1(M)>0$), curvature bounds ($\operatorname{Ric}\ge0$), decay assumptions on $f$, or a weighted Poincaré inequality at infinity (Catino et al., 2019).

A weighted Poincaré inequality at infinity asserts that for every $R\ge R_0$ there exists a nonnegative, locally bounded "weight" function $P_R: M \rightarrow [0,\infty)$ such that

$$\int_M P_R v^2\,dV \le \int_M |\nabla v|^2\,dV,\quad \forall v\in C_c^\infty(M\setminus B_R(p)),$$

and the conformal metric $g_{P_R} := P_R\,g$ is complete on $M\setminus B_R(p)$. This property, denoted $(\mathcal P_\infty)$, critically tames the behavior of the zero-mode of $-\Delta$ at infinity, independent of any spectral gap.

2. Ricci Curvature and Effects on Weights

Denoting the Ricci tensor by $\operatorname{Ric}$ and $p(x) = \min_{|v|=1} \operatorname{Ric}(v,v)(x)$, the framework assumes (outside large compact sets) lower bounds on Ricci curvature of the form

$$p(x) \ge -w(r(x)), \quad r(x) = \operatorname{dist}(x, p),$$

where $w: [0,\infty)\to (0,\infty)$ is continuous and nondecreasing. Model bounds typical in the literature are

$$\operatorname{Ric}(\nabla r, \nabla r) \ge -C(1+r)^\gamma,\quad (\gamma\in\mathbb{R}),$$

yielding $w(r)\sim (1+r)^\gamma$ or, for critical cases like $\gamma = -2$, $w(r)\sim\ln(1+r)$. These growth rates directly inform admissible choices of the weight functions $P_R$, interfacing naturally with Hardy-type inequalities and local spectral estimates (cf. Carron, Li–Wang, Berchio–Ganguly–Grillo) (Catino et al., 2019).

3. Main Existence and Uniqueness Theorems

General Weighted-Poincaré Case

Let $B_m(p)$ denote the geodesic ball of radius $m$ about $p$ and $P_m := P_{R=m}$. If $f\in C^\alpha_{\textrm{loc}}(M)$ and

$$\sum_{m=0}^\infty \left[w(m+1) - w(m) + 1\right] \sup_{x\in M\setminus B_m(p)} \frac{|f(x)|}{P_m(x)} < \infty,$$

then there exists a classical solution $u\in C^2(M)$ to $\Delta_g u = f$ (Catino et al., 2019).

Non-Parabolic Case with Hardy-Type Weight

If $M$ is non-parabolic (possessing a positive minimal Green’s function $G$) and $p(x) = |\nabla_x G(p,x)|^2 / 4G(p,x)^2$, then

$$\sum_{m=0}^\infty \left[w(m+1) - w(m)\right] \sup_{M\setminus B_m(p)} \frac{|f|}{p} < \infty$$

again ensures the existence of a classical solution.

This separating theorems from global lower Ricci bounds or spectral gap enables the extension of classical results to a much broader geometric setting (Catino et al., 2019).

4. Decay Thresholds and Examples

Under these frameworks, sharp pointwise decay rates on the source $f$ are both necessary and sufficient. In key cases:

• Euclidean space $\mathbb{R}^n$ ($n \ge 3$): $P_m(x)\simeq |x|^{-2}$, $w(r)\sim \ln r$, so

$|f(x)|\le C(1+|x|)^{-a},\ a>2$

is both necessary and sharp.

• Hyperbolic space $\mathbb{H}^n$: $P_m \sim 1$, $w(m+1) - w(m) \sim 1$, so $|f|\le C(1+r)^{-a},\ a>1$ is sufficient and necessary.
• Cartan–Hadamard manifolds with $\operatorname{Ric}\le -C(1+r)^\gamma$: $P_m\sim m^{-\gamma_2}$, reflecting the generalized result of Ni–Shi–Tam and Munteanu–Sesum (Catino et al., 2019).

A table summarizing principal decay rates:

Geometry	Weight $P_m$	Sharp $|f|$ Decay
$\mathbb{R}^n$	$|x|^{-2}$	$(1+|x|)^{-a},\ a>2$
$\mathbb{H}^n$	$1$	$(1+r)^{-a},\ a>1$
Hadamard ($\gamma$)	$m^{-\gamma_2}$	Model-dependent, see text

5. Green's Function Formulation and Proof Techniques

Potential solutions are expressed as Green integrals: $u(x) = \int_M G(x,y) f(y) dV(y).$ Green’s function $G(x,y)$ for the Laplacian or the weighted Laplacian (e.g., $$\Delta_{\rho} = \rho^{-1}\operatorname{Div}(\rho \nabla \cdot)$$) is minimal, symmetric, and positive.

Critical analytic steps include:

• Local Harnack and Cheng–Yau gradient estimates for Green’s function asymptotics.
• Co-area decomposition along $G$-level sets with $|\nabla_y G|$ equidistribution exploited using $$\int_{G=s} |\nabla_y G|\,d\mathcal{H}^{n-1} \equiv 1$$.
• Summation over dyadic annular shells, where the weighted Poincaré inequality quantifies each shell's contribution (Catino et al., 2019, Munteanu et al., 2019).

In the weighted setting, existence, uniqueness, and decay of solutions are controlled by integral bounds: $\int_{B_\rho(p,r)} \rho(y)\,G(x,y)\,d\Vol_g(y)\le C(r+1)$ and exponential off-diagonal decay: $G(x,z)\le Ce^{-a\,d_\rho(x,z)}$ under volume growth assumptions for the conformal metric $ds^2_\rho = \rho ds^2_g$ (Munteanu et al., 2019).

6. Generality, Limitations, and Recovery of Classical Results

No global Ricci curvature lower bound or global spectral gap is required—only quantitative inequalities at infinity. The approach recovers and unifies classical solvability conditions (e.g., Li–Tam–Ni–Shi) and is flexible enough to include Ricci-flat-at-infinity, negatively curved, or model geometries (Catino et al., 2019).

Key general and limiting considerations:

• The completeness of the conformal metric $g_{P_R}$ is critical—if $P_R$ decays too rapidly, the method fails.
• The decay assumption on $f$ relative to $P_R$ is sharp, as established via radially symmetric examples.
• The theory can be viewed through the lens of local spectral gaps or Hardy-type inequalities.
• Regularity hypotheses on $f$ (e.g., $C^\alpha_{\mathrm{loc}}$) are required for the Green representation and certain maximum principle arguments; these may be weakened by approximation (Catino et al., 2019).

7. Connections, Significance, and Research Context

The curvature-weighted Poisson equation theory, as developed by Catino–Monticelli–Punzo (Catino et al., 2019) and further analyzed by Munteanu–Sung–Wang (Munteanu et al., 2019), provides a unified analytic and geometric framework for solvability of Poisson and Laplace equations on non-compact manifolds. The shift from classical, global curvature or spectral assumptions to localized, weight-adapted inequalities reflects a general trend in geometric analysis toward finer quantification of geometric-degenerate regions and the influence of curvature on global analytical properties. This paradigm facilitates sharper, more general results, and has immediate consequences for classical analysis, geometric function theory, and global differential geometry.