Related papers: On the geometry of projective tensor products

We establish the necessary and sufficient conditions for constructing gradient Einstein-type warped metrics. One of these conditions leads us to a general Lichnerowicz equation with analytic and geometric coefficients for this class of…

Let $K \subset {\mathbb R}^2$ be an $o$-symmetric convex body, and $K^*$ its polar body. Then we have $|K|\cdot |K^*| \ge 8$, with equality if and only if $K$ is a parallelogram. ($| \cdot |$ denotes volume). If $K \subset {\mathbb R}^2$ is…

We consider $k$-dimensional central sections of the unit ball of $\ell_p^n$ (denoted $B_p^n$) and we prove that their volume are bounded by the volume of $B_p^n$ whenever $1<p<2$ and $1\le k\le (n-1)/2$ or $k=n-1$. We also consider $0<p<1$…

Let $X$ be a complex Banach space and $x,y\in X$. By definition, we say that $x$ is Birkhoff-James orthogonal to $y$ if $ \|x+\lambda y\|_{X} \geq \|x\|_{X}$ for all $\lambda \in \mathbb{C}$. We prove that $x$ is Birkhoff-James orthogonal…

We study Lagrangian immersions in the nearly K\"ahler $\mathbb{S}^6$ which are warped product manifolds of a $1$-dimensional base and a surface. Apart from the totally geodesic ones, they are either of constant sectional curvature…

We prove a relative volume comparison theorem of Petersen-Wei for both $L^P$-bound of Bakry-\'{E}mery Ricci curvature and gradient of potential function. As an application, we give a modified proof for a volume comparison and monotonicity…

On Kahler manifolds with Ricci curvature lower bound, assuming the real analyticity of the metric, we establish a sharp relative volume comparison theorem for small balls. The model spaces being compared to are complex space forms, i.e,…

We consider Laplacian eigenfunctions on a $d-$dimensional bounded domain $M$ (or a $d-$dimensional compact manifold $M$) with Dirichlet conditions. These operators give rise to a sequence of eigenfunctions $(e_\ell)_{\ell \in \mathbb{N}}$.…

Let $A$ and $B$ be Banach algebras and let $T$ be an algebra homomorphism from $B$ into $A$. The Cartesian product space $A\times B$ by $T$- Lau product and $\ell^{1}$- norm becomes a Banach algebra $A\times_{T}B$. We investigate the…

We consider the method of alternating (metric) projections for pairs of linear subspaces of finite dimensional Banach spaces. We investigate the size of the set of points for which this method converges to the metric projection onto the…

In this note we show the following result using the integral-geometric formula of R. Howard: Consider the totally geodesic $\mathbb{R}P^{2m}$ in $\mathbb{C}P^n$. Then it minimizes volume among the isotropic submanifolds in the same…

Consider a $q$-Weil polynomial $f$ of degree $2g$. Using an equidistribution assumption that is too strong to be true, we define and compute a product of local relative densities of matrices in $\rm{GSp}_{2g}(\mathbb{F}_\ell)$ with…

We prove that if a $\mathbb{Q}$-Fano variety $V$ specially degenerates to a K\"{a}hler-Einstein $\mathbb{Q}$-Fano variety $V$, then for any ample Cartier divisor $H=-r^{-1} K_V$ with $r\in \mathbb{Q}_{>0}$, the normalized volume…

We propose a new construction which associates to any ample (or big) line bundle $L$ on a projective manifold $X$ a canonical growth condition (i.e. a choice of a psh function well-defined up to a bounded term) on the tangent space $T_p X$…

We study the relationship between the tensor product multiplicities of a compact semisimple Lie algebra $\mathfrak{g}$ and a special function $\mathcal{J}$ associated to $\mathfrak{g}$, called the volume function. The volume function arises…

We consider hyperbolic links that admit alternating projections on surfaces in compact, irreducible 3-manifolds. We show that, under some mild hypotheses, the volume of the complement of such a link is bounded below in terms of a Kauffman…

We construct several new classes of bifunctors $(A,B)\mapsto A\otimes_{\alpha} B$, where $A\otimes_\alpha B$ is a cross norm completion of $A\odot B$ for each pair of C*-algebras $A$ and $B$. For the first class of bifunctors considered…

In recent years, sum-product estimates in Euclidean space and finite fields have been studied using a variety of combinatorial, number theoretic and analytic methods. Erdos type problems involving the distribution of distances, areas and…

We prove pointwise and $L^{p}$-gradient comparison results for solutions to elliptic Dirichlet problems defined on open subsets of a (possibly non-smooth) space with positive Ricci curvature (more precisely of an $\mathrm{RCD}(K,N)$ metric…

Given $E \subset {\Bbb R}^d$, define the \emph{volume set} of $E$, ${\mathcal V}(E)= \{det(x^1, x^2, ... x^d): x^j \in E\}$. In $\R^3$, we prove that ${\mathcal V}(E)$ has positive Lebesgue measure if either the Hausdorff dimension of…