Let $(M,\mathcal{D},g)$ be a smooth equiregular sub-Riemannian manifold equipped with a smooth positive measure $\mu$. We study the small-scale limit of the metric-ball mean-value operator \[ A_hf(x)=\frac{1}{h^{2}\mu(B(x,h)) }\int_{B(x,h)}(f(q)-f(x))\, d\mu(q). \] Exact homogeneity yields the pointwise limit on Carnot groups. On a general equiregular manifold, convergence for every smooth test function is equivalent to convergence of the rescaled horizontal first moments of metric balls in first-kind privileged coordinates. This criterion is independent of $\mu$; when it holds, the principal symbol is determined by the normalized second-moment tensor of the tangent unit ball, and the drift satisfies an explicit change-of-measure formula. In step at most two, we verify the criterion by combining a real-analytic finite-jet reduction with tame integration, which rules out oscillation of the normalized moments. We also compute the limit on Lie groups and prove unconditional distributional convergence of the volume-weighted operators on every equiregular manifold.
Let $(M^3,g)$ be a complete, connected, noncompact Riemannian three-manifold without boundary and with nonnegative sectional curvature. We prove that its scalar-curvature integral over geodesic balls, divided by the radius, has a limit. In the one-ended case, \[ \lim_{r\to\infty}\frac1r\int_{B_p(r)}\operatorname{Scal}\...
Let $(M^n,g_0)$ be a closed smooth Riemannian manifold, with $n\geqslant 3$. We prove short--time existence and uniqueness for the"critical"{\em Ricci--Bourguignon flow} \begin{equation} \partial_t g=-2\operatorname{Ric}_g+\frac{R_g}{n-1}g\,. \end{equation} Up to a constant rescaling of time, this is the {\em Schouten...
We construct a compact, equiregular, ideal sub-Riemannian manifold of step $2$ such that, for every smooth positive measure, the $\mathrm{MCP}(K,N)$ fails for all $K\in\mathbb{R}$ and $N \in (1,\infty)$. This shows that the real-analyticity assumption in the measure contraction theorem of Badreddine and Rifford in arXi...
Let $(M,g)$ be a smooth, closed, connected $d$-dimensional Riemannian manifold, and let $X_1,\ldots,X_n$ be i.i.d.\ with common law $f\,d\mathrm{vol}_g$, where $f\in C^4(M)$ is strictly positive. We derive a uniform intrinsic second-order expansion for symmetric three-vertex edge-indicator statistics supported on conne...
Let $(M^{n}, g)$ be a complete, simply connected Riemannian manifold without boundary, of dimension $n\ge3$, with curvature operator at least that of the unit sphere. We prove that $$\int_M {\rm scal}(x)\ d {\rm Vol}_x\le n(n-1)\omega_n,$$ where $\omega_n$ is the volume of the unit $n$-sphere. Equality holds if and onl...
For immersions of a closed $2h$-dimensional manifold $\Sigma$ into $\mathbb R^m$, we consider curvature energies that are invariant under ambient isometries and dilations, with the principal term $\int_{\Sigma} \big|\nabla^{(h-1)} \vec{\mathrm{I\!I}}\big|_g^2\,d\text{vol}_g$ and lower-order submanifold invariants. Foll...
Tian Lan· 1 citation
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