Mathematics - Differential Geometry Publications (50)


Mathematics - Differential Geometry Publications

In this paper, a metric with G$_2$ holonomy and slow rate of convergence to the cone metric is constructed on a ball inside the cone over the flag manifold. Read More

In this paper, we will study the Laplacian flow of closed G$_2$ structures on compact 7-manifolds. We will show that, if the scalar curvature remains bounded, then it is \kappa-noncollpasing. As a corollary, any blow up limit at finite time must have maximal volume growth rate. Read More

In this paper, we study biconservative submanifolds in $\Sn$ and $\Hn$ with parallel mean curvature vector field and co-dimension $2$. We obtain some necessary and sufficient conditions for such submanifolds to be conservative. In particular, we obtain a complete classification of $3$-dimensional biconservative submanifolds in $\Sp^4\times\R$ and $\Hy^4\times\R$ with nonzero parallel mean curvature vector field. Read More

We complete the classification of maximal representations of uniform complex hyperbolic lattices in Hermitian Lie groups by dealing with the exceptional groups ${\rm E}_6$ and ${\rm E}_7$. We prove that if $\rho$ is a maximal representation of a uniform complex hyperbolic lattice $\Gamma\subset{\rm SU}(1,n)$, $n>1$, in an exceptional Hermitian group $G$, then $n=2$ and $G={\rm E}_6$, and we describe completely the representation $\rho$. The case of classical Hermitian target groups was treated by Vincent Koziarz and the second named author (arxiv:1506. Read More

In this paper we explore a characterisation of relative equilibria in terms of sectional curvatures of the Jacobi-Maupertuis metric. We consider the planar $N$-body problem with an attractive $1/r^{\alpha}$ potential for general masses. Let $q(t)$ be a relative equilibria, we show that the sectional curvature is zero along $q(t)$, for a certain set of planes containing $\dot{q}(t)$, if and only if $\alpha=2$. Read More

In this paper we present the Ricci curvature on cell-complexes and show the Gauss-Bonnnet type theorem on graphs and 2-complex that decomposes closed surface. The defferential forms on a cell complex is defined as linear maps on chain complex, and Laplacian operates this defferential forms. Then we construct the Bochner-Weitzenb\"ock formula and define the Ricci curvature. Read More

In this note, we reveal that our solution of Demailly's strong openness conjecture implies a matrix version of the conjecture; our solutions of two conjectures of Demailly-Koll\'{a}r and Jonsson-Mustat\u{a} implies the truth of twisted versions of the strong openness conjecture; our optimal $L^{2}$ extension implies Berndtsson's positivity of vector bundles associated to holomorphic fibrations over a unit disc. Read More

Let M be a real Bott manifold with K\"{a}hler structure. Using Ishida characterization we give necessary and sufficient condition for the existence of the Spin-structure on M. In proof we use the technic developed in Popko, Szczepa\'{n}ski "Cohomological rigity of oriented Hantzsche-Wendt manifolds" (Adv. Read More

We present a construction of a certain invariant 2-form form on the (real) symplectic group. It is used to define a symplectic form on the quotient by a maximal torus and to lift a symplectic structure to the bundle of frames. Read More

We endow each closed, orientable Alexandrov space $(X, d)$ with an integral current $T$ of weight equal to 1, $\partial T = 0 and \set(T) = X$, in other words, we prove that $(X, d, T)$ is an integral current space with no boundary. Combining this result with a result of Li and Perales, we show that non-collapsing sequences of these spaces with uniform lower curvature and diameter bounds admit subsequences whose Gromov-Hausdorff and intrinsic flat limits agree. Read More

On a compact Riemannian manifold with boundary having positive mean curvature, a fundamental result of Shi and Tam states that, if the manifold has nonnegative scalar curvature and if the boundary is isometric to a strictly convex hypersurface in the Euclidean space, then the total mean curvature of the boundary is no greater than the total mean curvature of the corresponding Euclidean hypersurface. In $3$-dimension, Shi-Tam's result is known to be equivalent to the Riemannian positive mass theorem. In this paper, we provide a supplement to Shi-Tam's result by including the effect of minimal hypersurfaces on a chosen boundary component. Read More

We present a proof of Milnor conjecture in dimension 3 based on Cheeger-Colding theory on limit spaces of manifolds with Ricci curvature bounded below. It is different from [Liu] that relies on minimal surface theory. Read More

We prove that higher moment maps on area measures of a euclidean vector space are injective, while the kernel of the centroid map equals the image of the first variation map. Based on this, we introduce the space of smooth dual area measures on a finite-dimensional euclidean vector space and prove that it admits a natural convolution product which encodes the local additive kinematic formulas for groups acting transitively on the unit sphere. As an application of this new integral-geometric structure, we obtain the local additive kinematic formulas in hermitian vector spaces in a very explicit way. Read More

A 4-dimensional Riemannian manifold equipped with a circulant structure, which is an isometry with respect to the metric and its fourth power is the identity, is considered. The almost product manifold associated with the considered manifold is studied. The relation between the covariant derivatives of the almost product structure and the circulant structure is obtained. Read More

Let $X$ be a compact connected Riemann surface of genus at least two, and let ${\mathcal Q}_X(r,d)$ be the quot scheme that parametrizes all the torsion coherent quotients of ${\mathcal O}^{\oplus r}_X$ of degree $d$. This ${\mathcal Q}_X(r,d)$ is also a moduli space of vortices on $X$. Its geometric properties have been extensively studied. Read More

It is proved that the equality $\Delta\ln|\kappa-\lambda|=6\kappa$, where $\kappa$ is the Gaussian curvature of a metric tensor g on a 2-dimensional manifold is a sufficient and necessary condition for local realizability of the metric as the Blaschke metric of some affine sphere. Read More

The paper is focused on the existence problem of attractors for foliations. Since the existence of an attractor is a transversal property of the foliation, it is natural to consider foliations admitting transversal geometric structures. As transversal structures are chosen Cartan geometries due to their universality. Read More

We study the relationship between many natural conditions that one can put on a diffeological vector space, including being fine or projective, having enough smooth (or smooth linear) functions to separate points, having a diffeology determined by the smooth linear functionals, and more. Our main result is that the majority of the conditions fit into a total order. We also give many examples in order to show which implications do not hold. Read More

We obtain a Chern-Osserman type equality of a complete properly immersed surface in Euclidean space, provided the L^2-norm of the second fundamental form is finite. Also, by using a monotonicity formula, we prove that if the L^2-norm of mean curvature of a noncompact surface is finite, then it has at least quadratic area growth. Read More

We prove that certain involutions defined by Vogell and Burghelea-Fiedorowicz on the rational algebraic K-theory of spaces coincide. This gives a way to compute the positive and negative eigenspaces of the involution on rational homotopy groups of pseudoisotopy spaces from the involution on rational $S^{1}$--homology group of the free loop space of a simply-connected manifold. As an application, we give explicit dimensions of the open manifolds $V$ that appear in Belegradek-Farrell-Kapovitch's work for which the spaces of complete nonnegatively curved metrics on $V$ have nontrivial rational homotopy groups. Read More

We study generalized complex structures and $T$-duality (in the sense of Bouwknegt, Evslin, Hannabuss and Mathai) on Lie algebras and construct the corresponding Cavalcanti and Gualtieri map. Such construction is called {\em Infinitesimal $T$-duality}. As an application we deal with the problem of finding symplectic structures on 2-step nilpotent Lie algebras. Read More

We obtain Harnack estimates for a class of curvature flows in Riemannian manifolds of constant non-negative sectional curvature as well as in the Lorentzian Minkowski and de Sitter spaces. Furthermore, we prove a Harnack estimate with a bonus term for mean curvature flow in locally symmetric Riemannian Einstein manifold of non-negative sectional curvature. Using a concept of "duality" for strictly convex hypersurfaces, we also obtain a new type of inequalities, so-called "pseudo"-Harnack inequalities, for expanding flows in the sphere and in the hyperbolic space. Read More

Let M be a simple hyperkahler manifold. Kuga-Satake construction gives an embedding of H^2(M,C) into the second cohomology of a torus, compatible with the Hodge structure. We construct a torus T and an embedding of the graded cohomology space H^*(M,C) \to H^{*+l}(T,C) for some l, which is compatible with the Hodge structures and the Poincare pairing. Read More

In this paper we give formulae for the Dixmier trace and the Noncommutative Residue (also called Wodzicki's residue) for pseudo-differential operators by using the notion of global symbol in the setting of compact manifolds (with or without boundary). For the Dixmier trace --of invariant pseudo-differential operators-- in the case of compact manifolds, we use the Fourier analysis associated to every elliptic and positive operator. In particular, for compact Lie groups this can be done by using representation theory. Read More

Given a connected real Lie group and a contractible homogeneous proper $G$--space $X$ furnished with a $G$--invariant volume form, a real valued volume can be assigned to any representation $\rho\colon \pi_1(M)\to G$ for any oriented closed smooth manifold $M$ of the same dimension as $X$. Suppose that $G$ contains a closed and cocompact semisimple subgroup, it is shown in this paper that the set of volumes is finite for any given $M$. From a perspective of model geometries, examples are investigated and applications with mapping degrees are discussed. Read More

A singular (or Hermann) foliation on a smooth manifold $M$ can be seen as a subsheaf of the sheaf $\mathfrak{X}$ of vector fields on $M$. We show that if this singular foliation admits a resolution (in the sense of sheaves) consisting of sections of a graded vector bundle of finite type, then one can lift the Lie bracket of vector fields to a Lie $\infty$-algebroid structure on this resolution, that we call a universal Lie $\infty$-algebroid associated to the foliation. The name is justified because it is isomorphic (up to homotopy) to any other Lie $\infty$-algebroid structure built on any other resolution of the given singular foliation. Read More

We revisit the problem of extension of a Killing vector field in a spacetime which is solution to the Einstein equation with electromagnetic stress/energy tensor. This extension has been proven by Yu to be unique in the case of a Killing vector field which is normal to a bifurcate horizon. Here we generalize the extension of the vector field to a strong null convex domain in an electrovacuum spacetime, inspired by the same technique used by Ionescu and Klainerman in the setting of Ricci flat manifolds. Read More

We show that a Hitchin representation is determined by the spectral radii of the images of simple, non-separating closed curves. As a consequence, we classify isometries of the intersection function on Hitchin components of dimension 3 and on the self-dual Hitchin components in all dimensions. Read More

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 (2+1) spacetime. The proof follows from polyhedral approximation. Read More

We prove regularity and well-posedness results for the mixed Dirichlet-Neumann problem for a second order, uniformly strongly elliptic differential operator on a manifold $M$ with boundary $\partial M$ and bounded geometry. Our well-posedness result for the Laplacian $\Delta_g := d^*d \ge 0$ associated to the given metric require the additional assumption that the pair $(M, \partial_D M)$ be of finite width (in the sense that the distance to $\partial_D M$ is bounded uniformly on $M$, where $\partial_D M$ is the Dirichlet part of the boundary). The proof is a continuation of the ideas in our previous paper on the Dirichlet problem on manifolds with boundary and bounded geometry (joint with Bernd Ammann). Read More

In [3] and [11] the authors showed the existence of a Codazzi pair defined on any constant mean curvature surface in the homogeneous spaces E($\kappa$,$\tau$) associated to the Abresch-Rosenberg differential. In this paper, we use the mentioned Codazzi pair to classify capillary disks in E($\kappa$,$\tau$). As a consequence, the results presented in this paper generalize the previous classification of constant mean curvature disks in the product spaces $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ in [4] and [5]. Read More

We conjecture a formula for the generating function of virtual $\chi_y$-genera of moduli spaces of rank 2 sheaves on minimal surfaces of general type. Specializing this conjecture to virtual Euler characteristics, we recover (part of) a formula of C. Vafa and E. Read More

We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form $F^{-p}$, where $p>1$ and $F$ is a positive, strictly monotone, homogeneous of degree $1$ and concave curvature function. In particular this class includes the mean curvature $F=H$. Read More

In this paper, the dual Orlicz curvature measure is proposed and its basic properties are provided. A variational formula for the dual Orlicz-quermassintegral is established in order to give a geometric interpretation of the dual Orlicz curvature measure. Based on the established variational formula, a solution to the dual Orlicz-Minkowski problem regarding the dual Orlicz curvature measure is provided. Read More

In this essay we aim to explore the Geometric aspects of the Calabi Conjecture and highlight the techniques of nonlinear Elliptic PDE theory used by S.T. Yau [SY] in obtaining a solution to the problem. Read More

In 1981 Edward Witten proved a remarkable result where he derived the classical Morse Inequalities using ideas from Supersymmetric (SUSY) Quantum Mechanics. In this regard, one has an example where a Physical Theory has something to say about the underlying Mathematical Structure. The objective of this essay is to understand this classical result from the perspective of Schr\"{o}dinger Operators. Read More

We show that the action functional of the nonlinear sigma model with gravitino considered in a previous article [18] is invariant under rescaled conformal transformations, super Weyl transformations and diffeomorphisms. We give a careful geometric explanation how a variation of the metric leads to the corresponding variation of the spinors. In particular cases and despite using only commutative variables, the functional possesses a degenerate super symmetry. Read More

Let $M$ be a compact and connected smooth manifold endowed with a smooth action of a finite group $\Gamma$, and let $f$ be a $\Gamma$-invariant Morse function on $M$. We prove that the space of $\Gamma$-invariant Riemannian metrics on $M$ contains a residual subset ${\mathcal M}_f$ with the following property. Let $g\in{\mathcal M}_f$ and let $\nabla^gf$ be the gradient vector field of $f$ with respect to $g$. Read More

These are extended notes of the course given by the author at RIMS, Kyoto, in October 2016. The aim is to give a self-contained overview on the recently developed approach to differential calculus on metric measure spaces. The effort is directed into giving as many ideas as possible, without losing too much time in technical details and utmost generality: for this reason many statements are given under some simplifying assumptions and proofs are sometimes only sketched. Read More

We partially resolve a conjecture of Meeks on the asymptotic behavior of minimal surfaces in $\mathbb{R}^3$ with quadratic area growth. Read More

In this paper, we will give an Enneper-type representation for spacelike and timelike minimal surfaces in the Lorentz-Minkowski space $L^{3}$, using the complex and the paracomplex analysis (respectively). Then, we exhibit various examples of minimal surfaces in $L^{3}$ constructed via the Enneper representation formula, that it is equivalent to the Weierstrass representation obtained by Kobayashi (for spacelike immersions) and by Konderak (for the timelike ones). Read More

The covariant canonical formalism is a covariant extension of the traditional canonical formalism of fields. In contrast to the traditional canonical theory, it has a remarkable feature that canonical equations of gauge theories or gravity are not only manifestly Lorentz covariant but also gauge covariant or diffeomorphism covariant. A mathematical peculiarity of the covariant canonical formalism is that its canonical coordinates are differential forms on a manifold. Read More

We construct triply periodic zero mean curvature surfaces of mixed type in the Lorentz-Minkowski 3-space, with the same topology as the triply periodic minimal surfaces in the Euclidean 3-space, called Schwarz rPD surfaces. Read More

In this article, we consider a sequence solutions of Kapustin-Witten equations on a compact simply-connected four-manifold with general metric, we prove that when the anti-self-dual part of curvature converge to zero in $L^{2}$-topology, the extra fields converge to infinite in $L^{2}$-topology. Further more, we obtain that the curvatures of the non-trivial solutions with non-concentrating connections have a uniformly positive lower bounded in the sense of $L^{2}$. Read More

On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah-Singer Dirac operator $\mathrm{D}_{\mathcal B}$ in $\mathrm{L}^{2}$ depends Riesz continuously on $\mathrm{L}^{\infty}$ perturbations of local boundary conditions ${\mathcal B}$. The Lipschitz bound for the map ${\mathcal B} \to {\mathrm{D}}_{\mathcal B}(1 + {\mathrm{D}}_{\mathcal B}^2)^{-\frac{1}{2}}$ depends on Lipschitz smoothness and ellipticity of ${\mathcal B}$ and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions. Read More

This paper investigates gradient recovery schemes for data defined on discretized manifolds. The proposed method, parametric polynomial preserving recovery (PPPR), does not ask for the tangent spaces of the exact manifolds which have been assumed for some significant gradient recovery methods in the literature. Another advantage of the proposed method is that it removes the symmetric requirement from the existing methods for the superconvergence. Read More

We show that the concept of $H^2$-gradient flow for the Willmore energy and other functionals that depend at most quadratically on the second fundamental form is well-defined in the space of immersions of Sobolev class $W^{2,p}$ from a compact, $n$-dimensional manifold into Euclidean space, provided that $p \geq 2$ and $p>n$. We also discuss why this is not true for Sobolev class $H^2=W^{2,2}$. In the case of equality constraints, we provide sufficient conditions for the existence of the projected $H^2$-gradient flow and demonstrate its usability for optimization with several numerical examples. Read More

We derive a representation formula for the tensorial wave equation $\Box_\bg \phi^I=F^I$ in globally hyperbolic Lorentzian spacetimes $(\M^{2+1}, \bg)$ by giving a geometric formulation of the method of descent which is applicable for any dimension. Read More

We study the Berezin-Toeplitz quantization using as quantum space the space of eigenstates of the renormalized Bochner Laplacian corresponding to eigenvalues localized near the origin on a symplectic manifold. We show that this quantization has the correct semiclassical behavior and construct the corresponding star-product. Read More

We consider 3+1 rotationally symmetric Lorentzian Einstein spacetime manifolds with $\Lambda >0$ and reduce the equations to 2+1 Einstein equations coupled to `shifted' wave maps. Subsequently, we prove various (explicit) positive mass-energy theorems. No smallness is assumed. Read More