Rigidity of weighted manifolds via classification results for semilinear equations

We study model semilinear equations on complete and non-compact weighted Riemannian manifolds with non-negative Bakry-Émery Ricci curvature. Our main goal is to classify positive solutions of the equation at the Sobolev-critical exponent, and furthermore to prove that the existence of such solutions implies rigidity of the manifold and triviality of the weight. This is possible when the weighted manifold has non-negative finite dimensional Bakry-Émery Ricci curvature, and even under the weaker condition of non-negative infinite dimensional Bakry-Émery Ricci curvature, up to imposing some additional conditions in the latter case. To exhibit the sharpness of these additional conditions, we construct a non-trivial positive solution of the critical problem on a weighted manifold with positive infinite dimensional curvature. We also obtain a corresponding rigidity result for solutions of the Liouville equation on weighted Riemannian surfaces. Finally, we prove some non-existence theorems when the nonlinearity is sub-critical or simply under certain volume growth conditions. In particular, the latter rules out all positive solutions on shrinking gradient Ricci solitons.

💡 Research Summary

This paper establishes a series of rigidity and non-existence results for solutions to semilinear elliptic equations on complete, non-compact weighted Riemannian manifolds, linking the existence of specific solutions directly to the underlying geometry of the manifold and the triviality of the weight function.

The core object of study is the weighted Lane-Emden equation $-\mathcal{L} u = u^p$, where $\mathcal{L} = \Delta - \nabla f \cdot \nabla$ is the weighted Laplacian on a manifold $(M, g)$ with measure $d\mu = e^{-f}d\nu$. The curvature condition is given by the Bakry-Émery Ricci tensor $\text{Ric}_{n,d}$, which incorporates the Hessian of the weight $f$. The parameter $n \ge d$ acts as a “virtual dimension.”

The main results are structured around two primary curvature assumptions:

1. Under the infinite-dimensional curvature condition $\text{Ric}_{\infty,d} \ge 0$: The authors prove a rigidity theorem (Theorem 2.1) for positive solutions at the Sobolev-critical exponent $p = p_S(d) = (d+2)/(d-2)$. If such a solution exists and satisfies additional hypotheses—namely a Bishop-Gromov type volume growth condition (1.11) and the pointwise condition $\nabla f \cdot \nabla u \le 0$—then $(M,g)$ must be isometric to Euclidean space $\mathbb{R}^d$, the weight $f$ must be constant (so $\mathcal{L}=\Delta$), and $u$ must be an Aubin-Talenti bubble (1.3). The necessity of the condition $\nabla f \cdot \nabla u \le 0$ is dramatically highlighted by Theorem 2.4, where the authors construct explicit examples of weighted “model manifolds” with $\text{Ric}_{\infty,d} > 0$ that admit positive, bounded solutions to the critical equation for which $\nabla f \cdot \nabla u > 0$, thereby showing the rigidity theorem fails without this extra assumption.

2. Under the finite-dimensional curvature condition $\text{Ric}_{n,d} \ge 0$ ($n \ge d$): A cleaner rigidity result (Theorem 2.7) is obtained for solutions of $- \mathcal{L} u = u^{p_S(n)}$ at the exponent corresponding to the virtual dimension $n$. Here, under suitable asymptotic decay or finite energy assumptions on $u$ (but notably without requiring $\nabla f \cdot \nabla u \le 0$), the same conclusion holds: $(M,g)$ is Euclidean, $f$ is constant, and $u$ is a bubble.

Alongside these rigidity theorems, the paper provides powerful non-existence (Liouville-type) results:

• Theorem 2.5 states that under $\text{Ric}_{\infty,d} \ge 0$, condition (1.11), and $\nabla f \cdot \nabla u \le 0$, the only non-negative solution for subcritical exponents $1 < p < p_S(d)$ is the trivial one, $u \equiv 0$.
• More generally, Theorem 2.6 proves that without any curvature assumption, if the weighted volume growth of geodesic balls is sufficiently slow, specifically $\mu(B_R(o)) = o(R^{2p/(p-1)})$, then no positive solution exists for any $p>1$. This has immediate consequences, ruling out positive solutions on any shrinking gradient Ricci soliton (which have finite weighted volume) and on a broader class of weighted manifolds.

The paper also includes a corresponding rigidity theorem for solutions of the Liouville equation $-\mathcal{L} u = e^u$ on weighted surfaces ($d=2$).

Methodologically, the proofs adapt and extend the strategies developed in