Abstract:
Given a positive integer $n$ and a compact connected Riemann surface $X$, we prove that the symmetric product $S^n(X)$ admits a Kaehler form of nonnegative holomorphic bisectional curvature if and only if $\text{genus}(X) \leq 1$. If $n$ is greater than or equal to the gonality of $X$, we prove that $S^n(X)$ does not admit any Kaehler form of nonpositive holomorphic sectional curvature.

Abstract:
A classic result by Gromov and Lawson states that a Riemannian metric of non--negative scalar curvature on the Torus must be flat. The analogous rigidity result for the standard sphere was shown by Llarull. Later Goette and Semmelmann generalized it to locally symmetric spaces of compact type and nontrivial Euler characteristic. In this paper we improve the results by Llarull and Goette, Semmelmann. In fact we show that if $(M,g_0)$ is a locally symmetric space of compact type with $\chi (M)\neq 0$ and $g$ is a Riemannian metric on $M$ with $\mathrm{scal}_g\cdot g\geq \mathrm{scal}_0\cdot g_0$, then $g$ is a constant multiple of $g_0$. The previous results by Llarull and Goette, Semmelmann always needed the two inequalities $g\geq g_0$ and $\mathrm{scal}_g\geq \mathrm{scal}_0$ in order to conclude $g=g_0$. Moreover, if $(S^{2m},g_0)$ is the standard sphere, we improve this result further and show that any metric $g$ on $S^{2m}$ of scalar curvature $\mathrm{scal}_g\geq (2m-1)\mathrm{tr}_g(g_0)$ is a constant multiple of $g_0$.

Abstract:
We establish extremality of Riemannian metrics g with non-negative curvature operator on symmetric spaces M=G/K of compact type with rk(G)-rk(K)\le 1. Let g' be another metric with scalar curvature k', such that g'\ge g on 2-vectors. We show that k'\ge k everywhere on M implies k'=k. Under an additional condition on the Ricci curvature of g, k'\ge k even implies g'=g. We also study area-non-increasing spin maps onto such Riemannian manifolds.

Abstract:
Given an smooth function $K <0$ we prove a result by Berger, Kazhdan and others that in every conformal class there exists a metric which attains this function as its Gaussian curvature for a compact Riemann surface of genus $g>1$. We do so by minimizing an appropriate functional using elementary analysis. In particular for $K$ a negative constant, this provides an elementary proof of the uniformization theorem for compact Riemann surfaces of genus $g >1$.

Abstract:
In this note we show that every (real or complex) vector bundle over a compact rank one symmetric space carries, after taking the Whitney sum with a trivial bundle of su?ciently large rank, a metric with nonnegative sectional curvature. We also examine the case of complex vector bundles over other manifolds, and give upper bounds for the rank of the trivial bundle that is necessary to add when the base is a sphere.

Abstract:
We prove that all convolution products of pairs of continuous orbital measures in rank one, compact symmetric spaces are absolutely continuous and determine which convolution products are in $L^{2}$ (meaning, their density function is in $L^{2})$. Characterizations of the pairs whose convolution product is either absolutely continuous or in $L^2$ are given in terms of the dimensions of the corresponding double cosets. In particular, we prove that if $G/K$ is not $SU(2)/SO(2),$ then the convolution of any two regular orbital measures is in $L^{2}$, while in $SU(2)/SO(2)$ there are no pairs of orbital measures whose convolution product is in $L^{2}$.

Abstract:
A conformal metric $g$ with constant curvature one and finite conical singularities on a compact Riemann surface $\Sigma$ can be thought of as the pullback of the standard metric on the 2-sphere by a multi-valued locally univalent meromorphic function $f$ on $\Sigma\backslash \{{\rm singularities}\}$, called the {\it developing map} of the metric $g$. When the developing map $f$ of such a metric $g$ on the compact Riemann surface $\Sigma$ has reducible monodromy, we show that, up to some M{\" o}bius transformation on $f$, the logarithmic differential $d\,(\log\, f)$ of $f$ turns out to be an abelian differential of 3rd kind on $\Sigma$, which satisfies some properties and is called a {\it character 1-form of} $g$. Conversely, given such an abelian differential $\omega$ of 3rd kind satisfying the above properties, we prove that there exists a unique conformal metric $g$ on $\Sigma$ with constant curvature one and conical singularities such that one of its character 1-forms coincides with $\omega$. This provides new examples of conformal metrics on compact Riemann surfaces of constant curvature one and with singularities. Moreover, we prove that the developing map is a rational function for a conformal metric $g$ with constant curvature one and finite conical singularities with angles in $2\pi\,{\Bbb Z}_{>1}$ on the two-sphere.

Abstract:
It is known that principal orbits of Hermann actions on a symmetric space of non-compact type are curvature-adapted isoparametric submanifolds having no focal point of non-Euclidean type on the ideal boundary of the ambient symmetric space. In this paper, we investigate the mean curvature flows for such a curvature-adapted isoparametric submanifold and its focal submanifold. Concretely the investigation is performed by investigating the mean curvature flows for the lift of the submanifold to an infinite dimensional pseudo-Hilbert space through a pseudo-Riemannian submersion.

Abstract:
Let G/K be a non-compact, rank-one, Riemannian symmetric space and let G^C be the universal complexification of G. We prove that a holomorphically separable, G-equivariant Riemann domain over G^C / K^C is necessarily univalent, provided that G is not a covering of SL(2, R). As a consequence of the above statement one obtains a univalence result for holomorphically separable, G x K -equivariant Riemann domains over G^C. Here G x K acts on G^C by left and right translations. The proof of such results involves a detailed study of the G-invariant complex geometry of the quotient G^C / K^C, including a complete classification of all its Stein G-invariant subdomains.

Abstract:
Users of Heegaard Floer homology may be reassured to know that it can be made to conform exactly to the standard analytic pattern of Lagrangian Floer homology. This follows from the following remark, which we prove using an argument of J. Varouchas: the natural singular K\"ahler form $\sym^n(\omega)$ on the $n$th symmetric product of a K\"ahler curve $(\Sigma,\omega)$ admits a cohomologous smoothing to a K\"ahler form which equals $\sym^n(\omega)$ away from a chosen neighbourhood of the diagonal.