arXiv subjects

Ruojing Jiang

Publications and source records attributed to Ruojing Jiang.

7 recordsLinked to original sources

arXiv2026

Asymptotically geodesic hypersurfaces and the fundamental groups of hyperbolic manifolds

We consider closed hypersurfaces smoothly immersed in hyperbolic manifolds up to homotopy and commensurability. We prove that if a closed hyperbolic manifold $M$ contains a sequence of asymptotically geodesic hypersurfaces, then $\pi_1(M)$ is virtually special and hence linear over integers. If $M$ (dimension at least 3) is, in addition, arithmetic of type I, we constructs a sequence of hypersurfaces which are asymptotically geodesic (but not totally geodesic), strongly filling, and equidistributing in the Grassmann bundle over $M$. This partially answers a question of Al Assal--Lowe. As a corollary, for each cocompact arithmetic lattice $\Gamma$ of $SO(n+1,1)$ of type I, there exist infinitely many arithmetic and infinitely many non-arithmetic cocompact lattices $H$ of $SO(n,1)$ that admit monomorphisms into $\Gamma$ which do not extend to a Lie group homomorphism from $SO(n,1)$ into $SO(n+1,1)$.

math.GT↗

arXiv2025

Area bounds for constant mean curvature surfaces in hyperbolic 3-manifolds

On a finite-volume hyperbolic $3$-manifold, we establish an upper bound on the area of closed embedded surfaces with constant mean curvature at least one, depending on the mean curvature and the genus bounds. This area bound implies compactness for such surfaces. In particular, for Bryant surfaces with constant mean curvature equal to one, the area is bounded proportionally to the genus.

math.DG↗

arXiv2025

Volume comparison on finite-volume hyperbolic 3-manifolds

On finite-volume hyperbolic $3$-manifolds, we compare volumes of different metrics using the exponential convergence of Ricci-DeTurck flow toward the hyperbolic metric $h_0$. We prove that among metrics with scalar curvature bounded below by $-6$, $h_0$ minimizes the volume. Moreover, for metrics that are either uniformly $C^2$-close to $h_0$ or asymptotically cusped of order at least two, equality holds if and only if the metric is isometric to $h_0$.

math.DG↗

arXiv2025

On the stability of Ricci flow on hyperbolic 3-manifolds of finite volume

On a hyperbolic 3-manifold of finite volume, we prove that if the initial metric is sufficiently close to the hyperbolic metric $h_0$, then the normalized Ricci-DeTurck flow exists for all time and converges exponentially fast to $h_0$ in a weighted H\"older norm. A key ingredient of our approach is the application of interpolation theory. Furthermore, this result is a valuable tool for investigating minimal surface entropy, which quantifies the growth rate of the number of closed minimal surfaces in terms of genus. We explore this in [17].

math.DG↗

arXiv2025

Minimal surface entropy and applications of Ricci flow on finite-volume hyperbolic 3-manifolds

This paper studies minimal surface entropy (the exponential asymptotic growth of the number of minimal surfaces up to a given value of area) for negatively curved metrics on hyperbolic $3$-manifolds of finite volume, particularly its comparison to the hyperbolic minimal surface entropy in terms of sectional and scalar curvature. On one hand, for metrics that are bilipschitz equivalent to the hyperbolic metric and have sectional curvature bounded above by $-1$ and uniformly bounded below, we show that the entropy achieves its minimum if and only if the metric is hyperbolic. On the other hand, by analyzing the convergence rate of the Ricci flow toward the hyperbolic metric, we prove that among all metrics with scalar curvature bounded below by $-6$ and with non-positive sectional curvature on the cusps, the entropy is maximized at the hyperbolic metric, provided that it is infinitesimally rigid. Furthermore, if the metrics are uniformly $C^0$-close to the hyperbolic metric and asymptotically cusped, then the entropy associated with the Lebesgue measure is uniquely maximized at the hyperbolic metric.

math.DG↗

arXiv2022

Average area ratio and normalized total scalar curvature of hyperbolic n-manifolds

On closed hyperbolic manifolds of dimension $n\geq 3$, we review the definition of the average area ratio of a metric $h$ with $R_h\geq -n(n-1)$ relative to the hyperbolic metric $h_0$, and we prove that it attains the local minimum of one at $h_0$, which solves a local version of Gromov's conjecture. Additionally, we discuss the relation between the average area ratio and normalized total scalar curvature for hyperbolic $n$-manifolds, as well as its relation to the minimal surface entropy if $n$ is odd.

math.DG↗

arXiv2022

Counting essential minimal surfaces in closed negatively curved n-manifolds

For closed odd-dimensional manifolds with sectional curvature less or equal than -1, we define the minimal surface entropy that counts the number of surface subgroups. It attains the minimum if and only if the metric is hyperbolic. Moreover, we compute the entropy associated with other locally symmetric spaces and cusped hyperbolic 3-manifolds.

math.DG↗