Large $p$-Capacitary Invariants, Entropy, and Geometric Rank

We introduce large-$p$ asymptotic invariants associated with the $p$-capacity, the first $p$-eigenvalue, and the Maz'ya constant on connected complete noncompact Riemannian manifolds. For the two standard normalizations \[ p\,\operatorname{Cap}_p(\Omega)^{1/p} \quad\text{and}\quad (p-1)\operatorname{Cap}_p(\Omega)^{1/(p-1)}, \] we prove that their upper and lower limits are independent of the bounded smooth conductor $\Omega$. We denote the conductor-independent upper limit of the first normalization by $\mathcal C(M)$. When the second normalization converges to a positive limit, its logarithmic second-order coefficient is also conductor-independent. These invariants satisfy \[ \mathcal V(M)\geq \mathcal C(M)\geq \Lambda(M)=\mathcal M(M)\geq0. \] Under centered-ball isoperimetry or rotational symmetry, together with an eventual monotonicity assumption on the sphere-area-to-ball-volume ratio, all four invariants coincide with the volume entropy. This yields hyperbolic rigidity from either maximal $p$-spectral data or maximal $\mathcal C(M)$, as well as an almost-rigidity theorem under Ricci curvature and diameter bounds. For the universal cover of a closed negatively curved manifold, both capacitary normalizations converge to the volume entropy, which equals the topological entropy of the geodesic flow, without any centered-ball isoperimetric or rotational-symmetry assumption. For nonflat Hadamard manifolds with compact quotient satisfying either of the above geometric conditions, we obtain a second-order large-$p$ expansion whose logarithmic coefficient detects the geometric rank. Finally, sharp examples show that the general inequalities may be strict and that the first-order capacitary limit need not exist.