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preprint2017arXiv

The equivariant Minkowski problem in Minkowski space

The classical Minkowski problem in Minkowski space asks, for a positive function $ϕ$ on $\mathbb{H}^d$ , for a convex set $K$ in Minkowski space with $C^2$ space-like boundary $S$ , such that $ϕ(η)^{-1}$ is the Gauss--Kronecker curvature at the point with normal $η$ . Analogously to the Euclidean case, it is possible to formulate a weak version of this problem: given a Radon measure $μ$ on $\mathbb{H}^d$ the generalized Minkowski problem in Minkowski space asks for a convex subset $K$ such that the area measure of $K$ is $μ$ . In the present paper we look at an equivariant version of the problem: given a uniform lattice $Γ$ of isometries of $\mathbb{H}^d$ , given a $Γ$ invariant Radon measure $μ$ , given a isometry group $Γ_τ$ of Minkowski space, with $Γ$ as linear part, there exists a unique convex set with area measure $μ$ , invariant under the action of $Γ_τ$ . The proof uses a functional which is the covolume associated to every invariant convex set. This result translates as a solution of the Minkowski problem in flat space times with compact hyperbolic Cauchy surface. The uniqueness part, as well as regularity results, follow from properties of the Monge--Ampère equation. The existen

preprint2016arXiv

Ricci solitons on low-dimensional generalized symmetric spaces

We consider three- and four-dimensional pseudo-Riemannian generalized symmetric spaces, whose invariant metrics were explicitly described in [15]. While four-dimensional pseudo-Riemannian generalized symmetric spaces of types A, C and D are algebraic Ricci solitons, the ones of type B are not so. The Ricci soliton equation for their metrics yields a system of partial differential equations. Solving such system, we prove that almost all the four-dimensional pseudo-Riemannian generalized symmetric spaces of type B are Ricci solitons. These examples show some deep differences arising for the Ricci soliton equation between the Riemannian and the pseudo-Riemannian cases, as any homogeneous Riemannian Ricci soliton is algebraic [21]. We also investigate three-dimensional generalized symmetric spaces of any signature and prove that they are Ricci solitons.

preprint2016arXiv

A 1-parameter family of spherical CR uniformizations of the figure eight knot complement

We describe a simple fundamental domain for the holonomy group of the boundary unipotent spherical CR uniformization of the figure eight knot complement, and deduce that small deformations of that holonomy group (such that the boundary holonomy remains parabolic) also give a uniformization of the figure eight knot complement. Finally, we construct an explicit 1-parameter family of deformations of the boundary unipotent holonomy group such that the boundary holonomy is twist-parabolic. For small values of the twist of these parabolic elements, this produces a 1-parameter family of pairwise non-conjugate spherical CR uniformizations of the figure eight knot complement.

preprint2017arXiv

Subregular subalgebras and invariant generalized complex structures on Lie groups

We introduce the notion of a subregular subalgebra, which we believe is useful for classification of subalgebras of Lie algebras. We use it to construct a non-regular invariant generalized complex structure on a Lie group. As an illustration of the study of invariant generalized complex structures, we compute them all for the real forms of G2.

preprint2017arXiv

Stability of anti-canonically balanced metrics

We study the asymptotic behavior of quantized Ding functionals along Bergman geodesic rays and prove that the slope at infinity can be expressed in terms of Donaldson-Futaki invariants and Chow weights. Based on the slope formula, we introduce a new algebro-geometric stability on Fano manifolds and show that the existence of anti-canonically balanced metrics implies our stability. The relation between our stability and others is also discussed. As another application of the slope formula, we get the lower bound estimate on the Calabi like functionals on Fano manifolds.

preprint2016arXiv

Infinite time singularities of the Kähler-Ricci flow

We study the long-time behavior of the Kahler-Ricci flow on compact Kahler manifolds. We give an almost complete classification of the singularity type of the flow at infinity, depending only on the underlying complex structure. If the manifold is of intermediate Kodaira dimension and has semiample canonical bundle, so that it is fibered by Calabi-Yau varieties, we show that parabolic rescalings around any point on a smooth fiber converge smoothly to a unique limit, which is the product of a Ricci-flat metric on the fiber and of a flat metric on Euclidean space. An analogous result holds for collapsing limits of Ricci-flat Kahler metrics.

preprint2016arXiv

An $L^{2}$ -isolation theorem for Yang-Mills fields on Kähler surfaces

We prove an $L^{2}$ energy gap result for Yang-Mills connections on principal $G$ -bundles over compact Kähler surfaces with positive scalar curvature. We prove related results for compact simply-connected Calabi-Yau $2$ -folds.

preprint2016arXiv

Spin goups of super metrics and a Theorem of Rogers

We derive the canonical forms of super Riemannian metrics and the local isometry groups of such metrics. For certain super metrics we also compute the simply connected covering groups of the local isometry groups and interpret these as local spin groups of the super metric. Using a generalization of a Theorem of Rogers, which is itself one of the main results of this paper, we show that for super metrics we call body reducible, each such simply connected covering group G is a super Lie group with a conventional super Lie algebra as its corresponding super Lie algebra. We work exclusively in the $G^{\infty}$ category.

preprint2017arXiv

$n$ -transitivity of bisection groups of a Lie groupoid

The notion of $n$ -transitivity can be carried over from groups of diffeomorphisms on a manifold $M$ to groups of bisections of a Lie groupoid over $M$ . The main theorem states that the $n$ -transitivity is fulfilled for all $n\in\mathbb N$ by an arbitrary group of $C^r$ -bisections of a Lie groupoid $Γ$ of class $C^r$ , where $1\leq r\leqω$ , under mild conditions. For instance, the group of all bisections of any Lie groupoid and the group of all Lagrangian bisections of any symplectic groupoid are $n$ -transitive in the sense of this theorem. In particular, if $Γ$ is source connected for any arrow $γ\inΓ$ there is a bisection passing through $γ$ .

preprint2016arXiv

Erratum to: Almost complex structures in 6D with non-degenerate Nijenhuis tensors and large symmetry groups

We correct an error in the second part of Theorem 3 of our original paper (arXiv:1512.07161).

preprint2016arXiv

Multiplicity free quasi-Hamiltonian manifolds

A (quasi-)Hamiltonian manifold is called multiplicity free if all of its symplectic reductions are 0-dimensional. In this paper, we classify multiplicity free Hamiltonian actions for (twisted) loop groups or, equivalently, multiplicity free (twisted) quasi-Hamiltonian manifolds for simply connected compact Lie groups. As a result we recover old and find new examples of these structures.

preprint2017arXiv

Duflo Isomorphism and Chern-Weil Theory

We explain how the distributional index of a transversally elliptic operator on a principal G-manifold P that is obtained by lifting a Dirac operator on P/G can serve as a link between the Duflo isomorphism and Chern-Weil forms.

preprint2017arXiv

Equilibrium Locus of The Flow on Circular Networks of Cells

We perform a geometric study of the equilibrium locus of the flow that models the diffusion process over a circular network of cells. We prove that when considering the set of all possible values of the parameters, the equilibrium locus is a smooth manifold with corners, while for a given value of the parameters, it is an embedded smooth and connected curve. For different values of the parameters, the curves are all isomorphic. Moreover, we show how to build a homotopy between different curves obtained for different values of the parameter set. This procedure allows the efficient computation of the equilibrium point for each value of some first integral of the system. This point would have been otherwise difficult to be computed for higher dimensions. We illustrate this construction by some numerical experiments. Eventually, we show that when considering the parameters as inputs, one can easily bring the system asymptotically to any equilibrium point in the reachable set, which we also easily characterize.

preprint2014arXiv

Reflection groups in non-negative curvature

We provide an equivariant description/classification of all complete (compact or not) non-negatively curved manifolds M together with a co-compact action by a reflection group W, and moreover, classify such W. In particular, we show that the building blocks consist of the classical constant curvature models and generalized open books with non negatively curved bundle pages, and derive a corresponding splitting theorem for the universal cover.

preprint2017arXiv

The first terms in the expansion of the Bergman kernel in higher degrees

We establish the cancellation of the first $2j$ terms in the diagonal asymptotic expansion of the restriction to the $(0,2j)$ -forms of the Bergman kernel associated to the spin ${}^c$ Dirac operator on high tensor powers of a positive line bundle twisted by a (non necessarily holomorphic) complex vector bundle, over a compact Kähler manifold. Moreover, we give a local formula for the first and the second (non-zero) leading coefficients.

preprint2013arXiv

Generalized Lawson tori and Klein bottles

Using Takahashi theorem we propose an approach to extend known families of minimal tori in spheres. As an example, the well-known two-parametric family of Lawson tau-surfaces including tori and Klein bottles is extended to a three-parametric family of tori and Klein bottles minimally immersed in spheres. Extremal spectral properties of the metrics on these surfaces are investigated. These metrics include i) both metrics extremal for the first non-trivial eigenvalue on the torus, i.e. the metric on the Clifford torus and the metric on the equilateral torus and ii) the metric maximal for the first non-trivial eigenvalue on the Klein bottle.

preprint2016arXiv

Notes on relative normalizations of ruled surfaces in the three-dimensional Euclidean space

This paper deals with relative normalizations of skew ruled surfaces in the Euclidean space $\mathbb{E}^{3}$ . In section 2 we investigate some new formulae concerning the Pick invariant, the relative curvature, the relative mean curvature and the curvature of the relative metric of a relatively normalized ruled surface $\varPhi$ and in section 3 we introduce some special normalizations of it. All ruled surfaces and their corresponding normalizations that make $\varPhi$ an improper or a proper relative sphere are determined in section 4. In the last section we study ruled surfaces, which are \emph{centrally} normalized, i.e., their relative normals at each point lie on the corresponding central plane. Especially we study various properties of the Tchebychev vector field. We conclude the paper by the study of the central image of $\varPhi$ .

preprint2017arXiv

The $f$ -Stability Index of the Constant Weighted Mean Curvature Hypersurfaces in Gradient Ricci Solitons

In this paper, we prove that a noncompact complete hypersurface with finite weighted volume, weighted mean curvature vector bounded in norm, and isometrically immersed in a complete weighted manifold is proper. In addition, we obtain an estimate for $f$ -stability index of a constant weighted mean curvature hypersurface with finite weighted volume and isometrically immersed in a shrinking gradient Ricci soliton that admits at least one parallel field globally defined. For such hypersurface, we still give a necessary condition for equality to be achieved in the estimate obtained.

preprint2016arXiv

SYZ Mirror Symmetry for Dirichlet Branes

In this thesis, we study a class of special Lagrangian submanifolds of toric Calabi-Yau manifolds and construct their mirrors using some techniques developed in the SYZ programme. We present a justification on the conjecture on the mirror construction of D- branes in Aganagic-Vafa [2]. We apply the techniques employed in Chan-Lau-Leung [8] and Chan [6], which give the SYZ mirror construction for D-branes. Recently, Chan-Lau-Leung [8] has given an explicit mirror construction for toric Calabi-Yau manifolds as complex algebraic varieties. Inspired by the SYZ transformation of D-branes in Leung-Yau-Zaslow[29], Chan[6] has defined a generalised mirror construction for a larger class of Lagrangian subspaces, which is important to our situation. The results point towards a need for further quantum correction of the SYZ transform for Lagrangian subspaces in order to recover physicists mirror prediction of counting holomorphic discs in Ooguri-Vafa [34]. When compare to physicists prediction of the Aganagic-Vafa mirror branes, such naive SYZ transform fails to coincide with the prediction. The main reasons lies in the fact that the SYZ transformation does not capture open Gromov-Witten inva

preprint2017arXiv

K-energy on polarized compactifications of Lie groups

In this paper, we study Mabuchi's K-energy on a compactification M of a reductive Lie group G, which is a complexification of its maximal compact subgroup K. We give a criterion for the properness of K-energy on the space of K \times K-invariant Kahler potentials. In particular, it turns to give an alternative proof of Delcroix's theorem for the existence of Kahler-Einstein metrics in case of Fano manifolds M . We also study the existence of minimizers of K-energy for general Kahler classes of M.

preprint2016arXiv

Topological Sigma Models On Supermanifolds

This paper concerns constructing topological sigma models governing maps from semirigid super Riemann surfaces to general target supermanifolds. We define both the A model and B model in this general setup by defining suitable BRST operators and physical observables. Using supersymmetric localization, we express correlation functions in these theories as integrals over suitable supermanifolds. In the case of the A model, we obtain an integral over the supermoduli space of "superinstantons". The language of supergeometry is used extensively throughout this paper.

preprint2017arXiv

The gauge fixing theorem with applications to the Yang-Mills flow over Riemannian manifolds

In 1982, Uhlenbeck \cite {U2} established the well-known gauge fixing theorem, which has played a fundamental role for Yang-Mills theory. In this paper, we apply the idea of Uhlenbeck to establish a parabolic type of gauge fixing theorems for the Yang-Mills flow and prove existence of a weak solution of the Yang-Mills flow on a compact $n$ -dimensional manifold with initial value $A_0$ in $W^{1,n/2}(M)$ . When $n=4$ , we improve a key lemma of Uhlenbeck (Lemma 2.7 of \cite {U2}) to prove uniqueness of weak solutions of the Yang-Mills flow on a four dimensional manifold.

preprint2003arXiv

Ambient metric construction of Q-curvature in conformal and CR geometries

We give a geometric derivation of Branson's Q-curvature in terms of the ambient metric associated with conformal structures; it naturally follows from the ambient metric construction of conformally invariant operators and can be applied to a large class of invariant operators. This procedure can be also applied to CR geometry and gives a CR analog of the Q-curvature; it then turns out that the Q-curvature gives the coefficient of the logarithmic singularity of the Szego kernel of 3-dimensional CR manifolds.

preprint2017arXiv

On homogeneous Hermite-Lorentz spaces

We define naturally Hermite-Lorentz metrics on almost-complex manifolds as special case of pseudo-Riemannian metrics compatible with the almost complex structure. We study their isometry groups.

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