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Fuquan Fang

Publications and source records attributed to Fuquan Fang.

At least 19 recordsLinked to original sources
arXiv2023

Topological classification of Bazaikin spaces

Geometry of manifolds with positive sectional curvature has been a central object dates back to the beginning of Riemannian geometry. Up to homeomorphism, there are only finitely many examples of simply connected positively curved manifolds in all dimensions except in dimension 7 and 13, namely, the Aloff-Wallach spaces and the Eschenburg spaces in dimension 7, and the Bazaikin spaces in dimension 13. The topological classification modelled on the 7-dimensional examples has been carried out by Kreck-Stolz which leads to a complete topological classification for the Aloff-Wallach spaces. The main goal of this paper is to provide the topological classification of 13-dimensional manifolds modelled on the Bazaikin spaces.

math.DG↗
arXiv2021

Isoperimetric Bounds for Lower Order Eigenvalues

New isoperimetric inequalities for lower order eigenvalues of the Laplacian on closed hypersurfaces, of the biharmonic Steklov problems and of the Wentzell-Laplace on bounded domains in a Euclidean space are proven. Some open questions for further study are also proposed.

math.AP↗
arXiv2017

Dual submanifolds in rational homology spheres

Let $Σ$ be a simply connected rational homology sphere. A pair of disjoint closed submanifolds $M_+, M_-$ in $Σ$ are called dual to each other if the complement $Σ- M_+$ strongly homotopy retracts onto $M_-$ or vice-versa. In this paper we will give a complete answer of which integral triples $(n; m_+, m_-)$ can appear, where $n=dim Σ-1$, $m_+={codim}M_+ -1$ and $m_-={codim}M_- -1$.

math.DG↗
arXiv2016

Tits Geometry and Positive Curvature

There is a well known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns-Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least three and irreducible topological spherical buildings of rank at least three. We discover and exploit a rich structure of a (connected) chamber system of finite (Coxeter) type M associated with any polar action of cohomogeneity at least two on any simply connected closed positively curved manifold. Although this chamber system is typically not a Tits geometry of type M, we prove that in all cases but two that its universal Tits cover indeed is a building. We construct a topology on this universal cover making it into a compact spherical building in the sense of Burns and Spatzier. Using this structure we classify up to equivariant diffeomorphism all polar actions on (simply connected) positively curved manifolds of cohomogeneity at least two.

math.DG↗
arXiv2016

Rank three geometry and positive curvature

An axiomatic characterization of buildings of type $\CC_3$ due to Tits is used to prove that any cohomogeneity two polar action of type $\CC_3$ on a positively curved simply connected manifold is equivariantly diffeomorphic to a polar action on a rank one symmetric space. This includes two actions on the Cayley plane whose associated $\CC_3$ type geometry is not covered by a building.

math.DG↗
arXiv2015

An almost flat manifold with a cyclic or quaternionic holonomy group bounds

A long-standing conjecture of Farrell and Zdravkovska and independently S.~T.~Yau states that every almost flat manifold is the boundary of a compact manifold. This paper gives a simple proof of this conjecture when the holonomy group is cyclic or quaternionic. The proof is based on the interaction between flat bundles and involutions.

math.DG↗
arXiv2014

Reflection groups in non-negative curvature

We provide an equivariant description/classification of all complete (compact or not) non-negatively curved manifolds M together with a co-compact action by a reflection group W, and moreover, classify such W. In particular, we show that the building blocks consist of the classical constant curvature models and generalized open books with non negatively curved bundle pages, and derive a corresponding splitting theorem for the universal cover.

math.DG↗
arXiv2008

Non-singular solutions of normalized Ricci flow on noncompact manifolds of finite volume

The main result of this paper shows that, if $g(t)$ is a complete non-singular solution of the normalized Ricci flow on a noncompact 4-manifold $M$ of finite volume, then the Euler characteristic number $χ(M)\geq0$. Moreover, $χ(M)\neq 0$, there exist a sequence times $t_k\to\infty$, a double sequence of points $\{p_{k,l}\}_{l=1}^{N}$ and domains $\{U_{k,l}\}_{l=1}^{N}$ with $p_{k,l}\in U_{k,l}$ satisfying the followings: [(i)] $\dist_{g(t_k)}(p_{k,l_1},p_{k,l_2})\to\infty$ as $k\to\infty$, for any fixed $l_1\neq l_2$; [(ii)] for each $l$, $(U_{k,l},g(t_k),p_{k,l})$ converges in the $C_{loc}^\infty$ sense to a complete negative Einstein manifold $(M_{\infty,l},g_{\infty,l},p_{\infty,l})$ when $k\to\infty$; [(iii)] $\Vol_{g(t_{k})}(M\backslash\bigcup_{l=1}^{N}U_{k,l})\to0$ as $k\to\infty$.

math.DG↗
arXiv2008

Knots in Riemannian manifolds

In this paper we study submanifold with nonpositive extrinsic curvature in a positively curved manifold. Among other things we prove that, if $K\subset (S^n, g)$ is a totally geodesic submanifold in a Riemannian sphere with positive sectional curvature where $n\ge 5$, then $K$ is homeomorphic to $S^{n-2}$ and the fundamental group of the knot complement $π_1(S^n-K)\cong \Bbb Z$.

math.DG↗
arXiv2007

Complete gradient shrinking Ricci solitons have finite topological type

We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its Bakry-Émery Ricci tensor has a positive lower bound, and either of the following conditions: (i) the Ricci curvature is bounded from above; (ii) the Ricci curvature is bounded from below and injectivity radius is bounded away from zero. Moreover, a complete shrinking Ricci soliton has finite topological type if its scalar curvature is bounded.

math.DG↗
arXiv2007

Convergence of Kaehler-Ricci flow with integral curvature bound

Let $g(t)$, $t\in [0, +\infty)$, be a solution of the normalized Kähler-Ricci flow on a compact Kähler $n$-manifold $M$ with $c_{1}(M)>0$ and initial metric $g (0)\in 2πc_{1}(M)$. If there is a constant $C$ independent of $t$ such that $$ \int_{M}|Rm(g(t))|^{n}dv_{t}\leq C,$$ then, for any $t_{k}\to \infty$, a subsequence of $(M, g(t_{k}))$ converges to a compact orbifold $(X, h)$ with only finite many singular points $\{q_{j}\}$ in the Gromov-Hausdorff sense, where $h$ is a Kähler metric on $X\backslash \{q_{j}\}$ satisfying the Kähler-Ricci soliton equation, i.e. there is a smooth function $f$ such that $$Ric(h)-h=\nabla\bar{\nabla}f, {\rm and}\it \nabla \nabla f=\bar{\nabla} \bar{\nabla} f=0. $$

math.DG↗
arXiv2007

Maximum solutions of normalized Ricci flows on 4-manifolds

We consider maximum solution $g(t)$, $t\in [0, +\infty)$, to the normalized Ricci flow. Among other things, we prove that, if $(M, ω) $ is a smooth compact symplectic 4-manifold such that $b_2^+(M)>1$ and let $g(t),t\in[0,\infty)$, be a solution to (1.3) on $M$ whose Ricci curvature satisfies that $|\text{Ric}(g(t))|\leq 3$ and additionally $χ(M)=3 τ(M)>0$, then there exists an $m\in \mathbb{N}$, and a sequence of points $\{x_{j,k}\in M\}$, $j=1, ..., m$, satisfying that, by passing to a subsequence, $$(M, g(t_{k}+t), x_{1,k},..., x_{m,k}) \stackrel{d_{GH}}\longrightarrow (\coprod_{j=1}^m N_j, g_{\infty}, x_{1,\infty}, ...,, x_{m,\infty}),$$ $t\in [0, \infty)$, in the $m$-pointed Gromov-Hausdorff sense for any sequence $t_{k}\longrightarrow \infty$, where $(N_{j}, g_{\infty})$, $j=1,..., m$, are complete complex hyperbolic orbifolds of complex dimension 2 with at most finitely many isolated orbifold points. Moreover, the convergence is $C^{\infty}$ in the non-singular part of $\coprod_1^m N_{j}$ and $\text{Vol}_{g_{0}}(M)=\sum_{j=1}^{m}\text{Vol}_{g_{\infty}}(N_{j})$, where $χ(M)$ (resp. $τ(M)$) is the Euler characteristic (resp. signature) of $M$.

math.DG↗
arXiv2006

Non-singular solutions to the normalized Ricci flow equation

In this paper we study non-singular solutions of Ricci flow on a closed manifold of dimension at least 4. Amongst others we prove that, if M is a closed 4-manifold on which the normalized Ricci flow exists for all time t>0 with uniformly bounded sectional curvature, then the Euler characteristic $χ(M)\ge 0$. Moreover, the 4-manifold satisfies one of the following \noindent (i) M is a shrinking Ricci solition; \noindent (ii) M admits a positive rank F-structure; \noindent (iii) the Hitchin-Thorpe type inequality holds 2χ(M)\ge 3|τ(M)| where $χ(M)$ (resp. $τ(M)$) is the Euler characteristic (resp. signature) of M.

math.DG↗
arXiv2006

Collapsed 5-manifolds with pinched positive sectional curvature

Let M be a closed 5-manifold of pinched curvature 0<δ\le \text{sec}_M\le 1. We prove that M is homeomorphic to a spherical space form if M satisfies one of the following conditions: (i) δ=1/4 and the fundamental group is a non-cyclic group of order at least C, a constant. (ii) The center of the fundamental group has index at least w(δ), a constant depending on δ. (iii) The ratio of the volume and the maximal injectivity radius is less than ε(δ). (iv) The volume is less than ε(δ) and the fundamental group π_1(M) has a center of index at least w, a universal constant, and π_1(M) is either isomorphic to a spherical 5-space group or has an odd order.

math.DG↗
arXiv2006

Perelman's λfunctional and the Seiberg-Witten equations

In this paper we study the supremum of Perelman's λ-functional {λ}_M(g) on Riemannian 4-manifold M by using the Seiberg-Witten equations. We prove among others that, for a compact Kähler-Einstein complex surface (M, J, g_{0}) with negative scalar curvature, (i) If g_{1} is a Riemannian metric on M with λ_{M}(g_{1})= λ_{M}(g_{0}), then Vol_{g_{1}}(M)\geq Vol_{g_{0}}(M). Moreover, the equality holds if and only if g_{1} is also a Kähler-Einstein metric with negative scalar curvature. (ii) If g_{t}, t\in [-1,1], is a family of Einstein metrics on M with initial metric g_{0}, then g_{t} is a Kähler-Einstein metric with negative scalar curvature.

math.FA↗