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In this note we relate the geometric notion of fill radius with the fundamental group of the manifold. We prove: Suppose that a closed Riemannian manifold M satisfies the property that its universal cover has bounded fill radius. Then the fundamental group of M is virtually free. We explain the relevance of this theorem to some conjectures on positive isotropic curvature and 2-positive Ricci curvature.
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We define local indices for projective umbilics and godrons (also called cusps of Gauss) on generic smooth surfaces in projective 3-space. By means of these indices, we provide formulas that relate the algebraic numbers of those characteristic points on a surface (and on domains of the surface) with the Euler characteristic of that surface (resp. of those domains). These relations determine the possible coexistences of projective umbilics and godrons on the surface. Our study is based on a fundamental cubic form for which we provide a closed simple expression.
First we confirm a conjecture asserting that any compact Kahler manifold $N$ with $Ric^perp>0$ must be simply-connected by applying a new viscosity consideration to Whitneys comass of $(p, 0)$-forms. Secondly we prove the projectivity and the rational connectedness of a Kahler manifold of complex dimension $n$ under the condition $Ric_k>0$ (for some $kin {1, cdots, n}$, with $Ric_n$ being the Ricci curvature), generalizing a well-known result of Campana, and independently of Kollar-Miyaoka-Mori, for the Fano manifolds. The proof utilizes both the above comass consideration and a second variation consideration of cite{Ni-Zheng2}. Thirdly, motivated by $Ric^perp$ and the classical work of Calabi-Vesentini cite{CV}, we propose two new curvature notions. The cohomology vanishing $H^q(N, TN)={0}$ for any $1le qle n$ and a deformation rigidity result are obtained under these new curvature conditions. In particular they are verified for all classical Kahler C-spaces with $b_2=1$. The new conditions provide viable candidates for a curvature characterization of homogenous Kahler manifolds related to a generalized Hartshone conjecture.
For compact manifolds with infinite fundamental group we present sufficient topological or metric conditions ensuring the existence of two geometrically distinct closed geodesics. We also show how results about generic Riemannian metrics can be carried over to Finsler metrics.
We study the character variety of representations of the fundamental group of a closed surface of genus $ggeq2$ into the Lie group SO(n,n+1) using Higgs bundles. For each integer $0<dleq n(2g-2),$ we show there is a smooth connected component of the character variety which is diffeomorphic to the product of a certain vector bundle over a symmetric product of a Riemann surface with the vector space of holomorphic differentials of degree 2,4,...,2n-2. In particular, when d=n(2g-2), this recovers Hitchins parameterization of the Hitchin component. We also exhibit $2^{2g+1}-1$ additional connected components of the SO(n,n+1)-character variety and compute their topology. Moreover, representations in all of these new components cannot be continuously deformed to representations with compact Zariski closure. Using recent work of Guichard and Wienhard on positivity, it is shown that each of the representations which define singularities (i.e. those which are not irreducible) in these $2^{2g+1}-1$ connected components are positive Anosov representations.
Let $S$ be a closed surface of genus at least $2$. For each maximal representation $rho: pi_1(S)rightarrowmathsf{Sp}(4,mathbb{R})$ in one of the $2g-3$ exceptional connected components, we prove there is a unique conformal structure on the surface in which the corresponding equivariant harmonic map to the symmetric space $mathsf{Sp}(4,mathbb{R})/mathsf{U}(2)$ is a minimal immersion. Using a Higgs bundle parameterization of these components, we give a mapping class group invariant parameterization of such components as fiber bundles over Teichmuller space. Unlike Labouries recent results on Hitchin components, these bundles are not vector bundles.
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