We obtain effective estimates for the growth rate of the $L^2$-energy of harmonic functions on geodesic balls in complete simply connected non-positively curved Riemannian manifolds with pinched sectional curvature. Our study relies upon a double-sided Price inequality for harmonic functions. Finally, we apply this circle of ideas to study the analytical structure of a potential counterexample to the Singer conjecture in degree one.
A Riemannian manifold is called almost positively curved if the set of points for which all $2$-planes have positive sectional curvature is open and dense. We find three new examples of almost positively curved manifolds: $Sp(3)/Sp(1)^2$, and two circle quotients of $Sp(3)/Sp(1)^2$. We also show the quasi-positively curved metric of Tapp [26]} on $Sp(n+1)/Sp(n-1) Sp(1)$ is not almost positively curved if $n\geq 3$.
In this paper, we analyze the asymptotic behavior of $κ$-noncollapsed and positively curved steady Ricci solitons and prove that any $n$-dimensional $κ$-noncollapsed steady Kähler-Ricci soliton with non-negative sectional curvature must be flat.
A Finsler space $(M,F)$ is called flag-wise positively curved, if for any $x\in M$ and any tangent plane $\mathbf{P}\subset T_xM$, we can find a nonzero vector $y\in \mathbf{P}$, such that the flag curvature $K^F(x,y, \mathbf{P})>0$. Though compact positively curved spaces are very rare in both Riemannian and Finsler geometry, flag-wise positively curved metrics should be easy to be found. A generic Finslerian perturbation for a non-negatively curved homogeneous metric may have a big chance to produce flag-wise positively curved metrics. This observation leads our discovery of these metrics on…
In this note, we show that any $n$-dimensional $κ$-noncollapsed steady Kähler-Ricci soliton with nonnegative bisectional curvature must be flat. The result is an improvement to our former work in \cite{DZ2}.
We classify positively curved Alexandrov spaces of dimension 4 with an isometric circle action up to equivariant homeomorphism, subject to a certain additional condition on the infinitesimal geometry near fixed points which we conjecture is always satisfied. As a corollary, we also classify positively curved Riemannian orbifolds of dimension 4 with an isometric circle action.
A classical result of Milman roughly states that every Lipschitz function on $\mathbb{S}^n$ is almost constant on a sufficiently high-dimensional sphere $\mathbb{S}^m\subset \mathbb{S}^n$. In this paper we extend the result by proving that any Lipschitz function on a positively curved homogeneous space is almost constant on a high dimensional submanifold.
In this paper, we study the geometry of the manifolds of geodesics of a Zoll surface of positive Gauss curvature, show how these metrics induce Finsler metrics of constant flag curvature and give some explicit constructions.
We prove that any metric of non-positive curvature in the sense of Alexandrov on a compact surface can be isometrically embedded as a convex spacelike Cauchy surface in a flat spacetime of dimension (2+1). The proof follows from polyhedral approximation.
Inspired by a recent work of Grove-Petersen in [GP18], where the authors studied Alexandrov spaces with largest possible boundary. We study Alexandrov spaces with lower curvature bound 1 and with small boundary. When the radius of X is π/2, and the boundary has diameter π/2, we classify the total space X.
We extend two known existence results to simply connected manifolds with positive sectional curvature: we show that there exist pairs of simply connected positively-curved manifolds that are tangentially homotopy equivalent but not homeomorphic, and we deduce that an open manifold may admit a pair of non-homeomorphic simply connected and positively-curved souls. Examples of such pairs are given by explicit pairs of Eschenburg spaces. To deduce the second statement from the first, we extend our earlier work on the stable converse soul question and show that it has a positive answer for a class …
We classify four-dimensional shrinking Ricci solitons satisfying $Sec \geq \frac{1}{24} R$, where $Sec$ and $R$ denote the sectional and the scalar curvature, respectively. They are isometric to either $\mathbb{R}^{4}$ (and quotients), $\mathbb{S}^{4}$, $\mathbb{RP}^{4}$ or $\mathbb{CP}^{2}$ with their standard metrics.
In this paper, we introduce the flag-wise positively curved condition for Finsler spaces (the (FP) Condition), which means that in each tangent plane, we can find a flag pole in this plane such that the corresponding flag has positive flag curvature. Applying the Killing navigation technique, we find a list of compact coset spaces admitting non-negatively curved homogeneous Finsler metrics satisfying the (FP) Condition. Using a crucial technique we developed previously, we prove that most of these coset spaces cannot be endowed with positively curved homogeneous Finsler metrics. We also prove …
We show that the shortest closed geodesic on a 2-sphere with non-negative curvature has length bounded above by three times the diameter. We prove a new isoperimetric inequality for 2-spheres with pinched curvature; this allows us to improve our bound on the length of the shortest closed geodesic in the pinched curvature setting.
In this article, we study complete Type I ancient Ricci flows with positive sectional curvature. Our main results are as follows: in the complete and noncompact case, all such ancient solutions must be noncollapsed on all scales; in the closed case, if the dimension is even, then all such ancient solutions must be noncollapsed on all scales. This furthermore gives a complete classification for three-dimensional noncompact Type I ancient solutions without assuming the noncollapsing condition.
We completely characterize the sectional curvature of all of the $13$-dimensional Bazaikin spaces. In particular, we show that all Bazaikin spaces admit a quasi-positively curved Riemannian metric, and that, up to isometry, there is a unique Bazaikin space which is almost positively curved but not positively curved.
A Ricci surface is defined as a Riemannian surface $(M,g_M)$ whose Gauss curvature satisfies the differential equation $KΔK + g_M(dK,dK) + 4K^3=0$. Andrei Moroianu and Sergiu Moroianu proved that a Ricci surface with non-positive Gauss curvature admits locally a minimal immersion into $\mathbb{R}^3$. In this paper, we are interested in studying non-compact orientable Ricci surfaces with catenoidal ends. We use an analogue of the Weierstrass data to obtain some classification results for such Ricci surfaces. We also give an existence result for positive genus Ricci surfaces with catenoidal ends…
In this paper, we prove two rigidity results for non-positively curved homogeneous Finsler metrics. Our first main result yields an extension of Hu-Deng's well-known result proven for the Randers metrics. Indeed, we prove that every connected homogeneous Finsler space with non-positive flag curvature and isotropic S-curvature is Riemannian or locally Minkowskian. We extend the Szabó's rigidity theorem for Berwald surfaces and show that homogeneous isotropic Berwald metrics with non-positive flag curvature are Riemannian or locally Minkowskian. We prove that a homogeneous $(α, β)$-metrics has i…
In this paper, we prove that on every Finsler manifold $(M,\,F)$ with reversibility $λ$ and flag curvature $K$ satisfying $\left(\fracλ{λ+1}\right)^2<K\le 1$, there exist $[\frac{\dim M+1}{2}]$ closed geodesics. If the number of closed geodesics is finite, then there exist $[\frac{\dim M}{2}]$ non-hyperbolic closed geodesics. Moreover, there are 3 closed geodesics on $(M,\,F)$ satisfying the above pinching condition when $\dim M=3$.
We show that a compact manifold admitting a Killing foliation with positive transverse curvature fibers over finite quotients of spheres or weighted complex projective spaces, provided that the singular foliation defined by the closures of the leaves has maximal dimension. This result is obtained by deforming the foliation into a closed one while maintaining transverse geometric properties, which allows us to apply results from the Riemannian geometry of orbifolds to the space of leaves. We also show that the basic Euler characteristic is preserved by such deformations. Using this fact we prov…
一个具有双不变正交分解$\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$的紧致黎曼齐性空间$G/H$称为交换对正弯曲的，如果对于$\mathfrak{m}$中线性无关交换对所张成的$T_{eH}(G/H)$中的任意切面，截面曲率为零.本文证明了在陪集空间$\mathrm{Sp}(2)/\mathrm{U}(1)$上，$\mathrm{U}(1)$对应于一个短根，交换对允许正曲度量. B. Wilking最近证明了这个$\mathrm{Sp}(2)/\mathrm{U}(1)$不可能是一般意义上的正弯曲。这是第一个例子，以区分一组紧凑的陪集空间承认正弯曲的度量，和度量正弯曲的交换对。
