Mathematics - Differential Geometry Publications (50)


Mathematics - Differential Geometry Publications

In this paper we investigate the spectral sequence associated to a Riemannian foliation which arises naturally on a Vaisman manifold. Using the Betti numbers of the underlying manifold we establish a lower bound for the dimension of some terms of this cohomological object. This way we obtain cohomological obstructions for two-dimensional foliations to be induced from a Vaisman structure. Read More

We prove (and improve) the Muir-Suffridge conjecture for holomorphic convex maps. Namely, let $F:\mathbb B^n\to \mathbb C^n$ be a univalent map from the unit ball whose image $D$ is convex. Let $\mathcal S\subset \partial \mathbb B^n$ be the set of points $\xi$ such that $\lim_{z\to \xi}\|F(z)\|=\infty$. Read More

We continue our study of geometric analysis on (possibly non-reversible) Finsler manifolds, based on the Bochner inequality established by the author and Sturm. Following the approach of the $\Gamma$-calculus a la Bakry et al, we show the dimensional versions of the Poincare--Lichnerowicz inequality, the logarithmic Sobolev inequality, and the Sobolev inequality. In the reversible case, these inequalities were obtained by Cavalletti--Mondino in the framework of curvature-dimension condition by means of the localization method. Read More

For each integer $k \geq 2$, we apply gluing methods to construct sequences of minimal surfaces embedded in the round $3$-sphere. We produce two types of sequences, all desingularizing collections of intersecting Clifford tori. Sequences of the first type converge to a collection of $k$ Clifford tori intersecting with maximal symmetry along these two circles. Read More

We first study holomorphic isometries from the Poincar\'e disk into the product of the unit disk and the complex unit $n$-ball for $n\ge 2$. On the other hand, we observe that there exists a holomorphic isometry from the product of the unit disk and the complex unit $n$-ball into any irreducible bounded symmetric domain of rank $\ge 2$ which is not biholomorphic to any type-$\mathrm{IV}$ domain. In particular, our study provides many new examples of holomorphic isometries from the Poincar\'e disk into irreducible bounded symmetric domains of rank at least $2$ except for type-$\mathrm{IV}$ domains. Read More

We study local ($n+1$)-webs of codimension 1 on a manifold of dimension $n.$ We give a complete description of their possible Lie algebras of infinitesimal diffeomorphisms. More precisely we show that these Lie algebras are direct products of sub-algebras which are isomorphic to $\frak{sl}(2),$ to the non-commutative 2-dimensional Lie algebra or commutative. Read More

We show that if a compact connected $n$-dimensional manifold $M$ has a conformal class containing two non-homothetic metrics $g$ and $\tilde g=e^{2\varphi}g$ with non-generic holonomy, then after passing to a finite covering, either $n=4$ and $(M,g,\tilde g)$ is an ambik\"ahler manifold, or $n\ge 6$ is even and $(M,g,\tilde g)$ is obtained by the Calabi Ansatz from a polarized Hodge manifold of dimension $n-2$, or both $g$ and $\tilde g$ have reducible holonomy, $M$ is locally diffeomorphic to a product $M_1\times M_2\times M_3$, the metrics $g$ and $\tilde g$ can be written as $g=g_1+g_2+e^{-2\varphi}g_3$ and $\tilde g=e^{2\varphi}(g_1+g_2)+g_3$ for some Riemannian metrics $g_i$ on $M_i$, and $\varphi$ is the pull-back of a non-constant function on $M_2$. Read More

Let M be a smooth compact manifold without boundary. We consider two smooth Sub-Semi-Riemannian metrics on M. Under suitable conditions, we show that they are almost conformally isometric in an Lp sense. Read More

In this work we study properties of stability and non-stability of harmonic maps under the homogeneous Ricci flow. We provide examples where the stability (non-stability) is preserved under the Ricci flow and an example where the Ricci flow does not preserve the stability of an harmonic map. Read More

We prove a sharp pinching estimate for immersed mean convex solutions of mean curvature flow which unifies and improves all previously known pinching estimates, including the umbilic estimate of Huisken, the convexity estimates of Huisken--Sinestrari and the cylindrical estimate of Huisken--Sinestrari. Namely, we show that the curvature of the solution pinches onto the convex cone generated by the curvatures of any shrinking cylinder solutions admitted by the initial data. For example, if the initial data is $(m+1)$-convex, then the curvature of the solution pinches onto the convex hull of the curvatures of the shrinking cylinders $\mathbb{R}^m\times S^{n-m}_{\sqrt{2(n-m)(1-t)}}$, $t<1$. Read More

Recently, at least 50 million of novel examples of compact $G_2$ holonomy manifolds have been constructed as twisted connected sums of asymptotically cylindrical Calabi-Yau threefolds. The purpose of this paper is to study mirror symmetry for compactifications of Type II superstrings in this context. We focus on $G_2$ manifolds obtained from building blocks constructed from dual pairs of tops, which are the closest to toric CY hypersurfaces, and formulate the analogue of the Batyrev mirror map for this class of $G_2$ holonomy manifolds, thus obtaining several millions of novel dual superstring backgrounds. Read More

We consider expanding vacuum spacetimes with a CMC foliation by compact spacelike hypersurfaces. Under scale invariant a priori geometric bounds (type-III), we show that there are arbitrarily large future time intervals that are modelled by a flat spacetime or a Kasner spacetime. We give related results for a class of expanding vacuum spacetimes that do not satisfy the a priori bounds (type-II). Read More

Let $Y$ be a closed Calabi-Yau manifold. Let $\omega$ be the K\"ahler form of a Ricci-flat K\"ahler metric on $\mathbb{C}^m \times Y$. We prove that if $\omega$ is uniformly bounded above and below by constant multiples of $\omega_{\mathbb{C}^m} + \omega_Y$, where $\omega_{\mathbb{C}^m}$ is the standard flat K\"ahler form on $\mathbb{C}^m$ and $\omega_Y$ is any K\"ahler form on $Y$, then $\omega$ is actually equal to a product K\"ahler form, up to a certain automorphism of $\mathbb{C}^m \times Y$. Read More

We prove that a theorem of Pawlucki, showing that Whitney regularity for a subanalytic set with a smooth singular locus of codimension one implies the set is a finite union of differentiable manifolds with boundary, applies to definable sets in polynomially bounded o-minimal structures. We give a refined version of Pawlucki's theorem for arbitrary o-minimal structures, replacing Whitney (b)-regularity by a quantified version, and we prove related results concerning normal cones and continuity of the density. We analyse two counterexamples to the extension of Pawlucki's theorem to definable sets in general o-minimal structures, and to several other statements valid for subanalytic sets. Read More

We prove integral curvature bounds in terms of the Betti numbers for compact submanifolds of the Euclidean space with low codimension. As an application, we obtain topological obstructions for $\delta$-pinched immersions. Furthermore, we obtain intrinsic obstructions for minimal submanifolds in spheres with pinched second fundamental form. Read More

The goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. More precisely, we give a new notion of angle between two curves in a metric space. Such a notion has a natural interplay with optimal transportation and is particularly well suited for metric measure spaces satisfying the curvature-dimension condition. Read More

This paper is a contribution to semiclassical analysis for abstract Schr\"odinger type operators on locally compact spaces: Let $X$ be a metrizable seperable locally compact space, let $\mu$ be a Radon measure on $X$ with a full support. Let $(t,x,y)\mapsto p(t,x,y)$ be a strictly positive pointwise consistent $\mu$-heat kernel, and assume that the generator $H_p\geq 0$ of the corresponding self-adjoint contraction semigroup in $L^2(X,\mu)$ induces a regular Dirichlet form. Then, given a function $\Psi : (0,1)\to (0,\infty)$ such that the limit $\lim_{t\to 0+}p(t,x,x)\Psi (t)$ exists for all $x\in X$, we prove that for every potential $w:X\to \mathbb{R}$ one has $$ \lim_{t \to 0+} \Psi (t)\mathrm{tr}\big(\mathrm{e}^{ -t H_p + w}\big)= \int \mathrm{e}^{-w(x) }\lim_{t \to 0+}p(t,x,x) \Psi (t) d\mu(x)<\infty $$ for the Schr\"odinger type operator $H_p + w$, provided $w$ satisfies very mild conditions at $\infty$, that are essentially only made to guarantee that the sum of quadratic forms $ H_p + w/t$ is self-adjoint and bounded from below for small $t$, and to guarantee that $$ \int \mathrm{e}^{-w(x) }\lim_{t\to 0+}p(t,x,x) \Psi (t) d\mu(x)<\infty. Read More

In this paper we characterize certain class of almost Kenmotsu metrics satisfying the Miao-Tam equation. Read More

A diffeological connection on a diffeological vector pseudo-bundle is defined just the usual one on a smooth vector bundle; this is possible to do, because there is a standard diffeological counterpart of the cotangent bundle. On the other hand, there is not yet a standard theory of tangent bundles, although there are many suggested and promising versions, such as that of the internal tangent bundle, so the abstract notion of a connection on a diffeological vector pseudo-bundle does not automatically provide a counterpart notion for Levi-Civita connections. In this paper we consider the dual of the just-mentioned counterpart of the cotangent bundle in place of the tangent bundle (without making any claim about its geometrical meaning). Read More

In this paper we provide a pinching condition for the characterization of the totally geodesic disk and the rotational annulus among minimal surfaces with free boundary in geodesic balls of three-dimensional hyperbolic space and hemisphere. The pinching condition involves the length of the second fundamental form, the support function of the surface, and a natural potential function in hyperbolic space and hemisphere. Read More

In this note, we prove the existence of weak solutions of the Chern-Ricci flow through blow downs of exceptional curves, as well as backwards smooth convergence away from the exceptional curves on compact complex surfaces. The smoothing property for the Chern-Ricci flow is also obtained on compact Hermitian manifolds of dimension n under a mild assumption. Read More

Using the weak solution of Inverse mean curvature flow, we prove the sharp Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space. Read More

This monograph was developed to support a series of lectures at the Institute of Mathematics at the Polish Academy of Sciences in September 2016, as part of a Workshop on the Geometry of Lagrangian Grassmannians and Nonlinear PDEs. The goal is to cut the shortest-possible expository path from the common, elementary concepts of geometry (linear algebra, vector bundles, and algebraic ideals) to the advanced theorems about the characteristic variety. Hopefully, these lectures lower the barrier to advanced topics in exterior differential systems by exposing the audience to elementary versions of several key results regarding the characteristic variety, and to outline how these results could be used to push the frontiers of the subject. Read More

To any $\mathfrak{g}$-manifold $M$ are associated two dglas $\operatorname{tot}\big(\Lambda^{\bullet} \mathfrak{g}^\vee \otimes_{\Bbbk} T_{\operatorname{poly}}^{\bullet} \big)$ and $\operatorname{tot} \big(\Lambda^{\bullet} \mathfrak{g}^\vee\otimes_{\Bbbk} D_{\operatorname{poly}}^{\bullet} \big)$, whose cohomologies $H_{\operatorname{CE}}(\mathfrak{g}, T_{\operatorname{poly}}^{\bullet} \xrightarrow{0} T_{\operatorname{poly}}^{\bullet+1})$ and $H_{\operatorname{CE}}(\mathfrak{g}, D_{\operatorname{poly}}^{\bullet} \xrightarrow{0} D_{\operatorname{poly}}^{\bullet+1})$ are Gerstenhaber algebras. We establish a formality theorem for $\mathfrak{g}$-manifolds: there exists an $L_\infty$ quasi-isomorphism $\Phi: \operatorname{tot}\big(\Lambda^{\bullet} \mathfrak{g}^\vee \otimes_{\Bbbk} T_{\operatorname{poly}}^{\bullet} \big) \to \operatorname{tot} \big(\Lambda^{\bullet} \mathfrak{g}^\vee\otimes_{\Bbbk} D_{\operatorname{poly}}^{\bullet} \big)$ whose first `Taylor coefficient' (1) is equal to the Hochschild-Kostant-Rosenberg map twisted by the square root of the Todd cocycle of the $\mathfrak{g}$-manifold $M$ and (2) induces an isomorphism of Gerstenhaber algebras on the level of cohomology. Consequently, the Hochschild-Kostant-Rosenberg map twisted by the square root of the Todd class of the $\mathfrak{g}$-manifold $M$ is an isomorphism of Gerstenhaber algebras from $H_{\operatorname{CE}}(\mathfrak{g}, T_{\operatorname{poly}}^{\bullet} \xrightarrow{0} T_{\operatorname{poly}}^{\bullet+1})$ to $H_{\operatorname{CE}}(\mathfrak{g}, D_{\operatorname{poly}}^{\bullet} \xrightarrow{0} D_{\operatorname{poly}}^{\bullet+1})$. Read More

On an asymptotically flat manifold $M^n$ with nonnegative scalar curvature, with outer minimizing boundary $\Sigma$, we prove a Penrose-like inequality in dimensions $ n < 8$, under suitable assumptions on the mean curvature and the scalar curvature of $ \Sigma$. Read More

Given a K\"ahler manifold $(Z,J,\omega)$ and a compact real submanifold $M\subset Z$, we study the properties of the gradient map associated with the action of a noncompact real reductive Lie group ${\rm G}$ on the space of probability measures on $M.$ In particular, we prove convexity results for such map when ${\rm G}$ is Abelian and we investigate how to extend them to the non-Abelian case. Read More

We provide a characterization of quotient spaces of three-dimensional complex tori by finite groups that act freely in codimension two via a vanishing condition on the first and second Chern class. For this we need to define the Chern classes of a complex space with canonical singularities, a notion which to the best of our knowledge has not been discussed in the literature before. Read More

As a generalization of anti-invariant $\xi^\perp-$Riemannian submersions, we introduce conformal anti-invariant $\xi^\perp-$submersions from almost contact metric manifolds onto Riemannian manifolds. We investigate the geometry of foliations which are arisen from the definition of a conformal submersion and find necessary and sufficient conditions for a conformal anti-invariant $\xi^\perp-$submersion to be totally geodesic and harmonic, respectively. Moreover, we show that there are certain product structures on the total space of a conformal anti-invariant $\xi^\perp-$submersion. Read More

Given a fiber bundle $Z \to M \to B$ and a flat vector bundle $E \to M$ with a compatible action of a discrete group $G$, and regarding $B / G$ as the non-commutative space corresponding to the crossed product algebra, we construct an analytic torsion form as a non-commutative deRham differential form. We show that our construction is well defined under the weaker assumption of positive Novikov-Shubin invariant. We prove that this torsion form appears in a transgression formula, from which a non-commutative Riamannian-Roch-Grothendieck index formula follows. Read More

We discuss the question of geometric formality for rationally elliptic manifolds of dimension $6$ and $7$. We prove that a geometrically formal six-dimensional biquotient with $b_{2}=3$ has the real cohomology of a symmetric space. We also show that a rationally hyperbolic six-dimensional manifold with $b_2\leq 2$ and $b_3=0$ can not be geometrically formal. Read More

Let $M$ be an open Riemann surface and $n\ge 3$ be an integer. We prove that on any closed discrete subset of $M$ one can prescribe the values of a conformal minimal immersion $M\to\mathbb{R}^n$. Our result also ensures jet-interpolation of given finite order, and hence, in particular, one may in addition prescribe the values of the generalized Gauss map. Read More

We show that pseudo-Riemannian almost quaternionic homogeneous spaces with index 4 and an H-irreducible isotropy group are locally isometric to a pseudo-Riemannian quaternionic K\"ahler symmetric space if the dimension is at least 16. In dimension 12 we give a non-symmetric example. Read More

Symmetry algebras of Killing vector fields and conformal Killing vectors fields can be extended to Killing-Yano and conformal Killing-Yano superalgebras in constant curvature manifolds. By defining $\mathbb{Z}$-gradations and filtrations of these superalgebras, we show that the second cohomology groups of them are trivial and they cannot be deformed to other Lie superalgebras. This shows the rigidity of Killing-Yano and conformal Killing-Yano superalgebras and reveals the fact that they correspond to geometric invariants of constant curvature manifolds. Read More

A vector field $B$ is said to be Beltrami vector field (force free-magnetic vector field in physics), if $B\times(\nabla\times B)=0$. Motivated by our investigations on projective superflows, and as an important side result, we construct two unique Beltrami vector fields $\mathfrak{I}$ and $\mathfrak{Y}$, such that $\nabla\times\mathfrak{I}=\mathfrak{I}$, $\nabla\times\mathfrak{Y}=\mathfrak{Y}$, and that both have orientation-preserving icosahedral symmetry (group of order $60$). Analogous constructions are done for tetrahedral and octahedral groups of orders $12$ and $24$, respectively. Read More

We study the question of the existence of left-invariant Sasaki contact structures on the seven-dimensional nilpotent Lie groups. It is shown that the only Lie group allowing Sasaki structure with a positive definite metric tensor is the Heisenberg group. We find a complete list of the 22 classes of seven-dimensional nilpotent Lie groups which admit pseudo-Sasaki structure. Read More

Let M be a compact Riemannian manifold with boundary. Let b>0 be the number of connected components of its boundary. For manifolds of dimension at least 3, we prove that it is possible to obtain an arbitrarily large (b+1)-th Steklov eigenvalue using a smooth conformal perturbation which is supported in a thin neighbourhood of the boundary, identically equal to 1 on the boundary. Read More

The twistor construction for Riemannian manifolds is extended to the case of manifolds endowed with generalized metrics (in the sense of generalized geometry \`a la Hitchin). The generalized twistor space associated to such a manifold is defined as the bundle of generalized complex structures on the tangent spaces of the manifold compatible with the given generalized metric. This space admits natural generalized almost complex structures whose integrability conditions are found in the paper. Read More

We prove that a general complex Monge-Amp\`ere flow on a Hermitian manifold can be run from an arbitrary initial condition with zero Lelong number at all points. Using this property, we confirm a conjecture of Tosatti-Weinkove: the Chern-Ricci flow performs a canonical surgical contraction. Finally, we study a generalization of the Chern-Ricci flow on compact Hermitian manifolds, namely the twisted Chern-Ricci flow. Read More

On Fano manifolds, the critical points of Ding energy is the Mabuchi metrics which Ricci potential defines a holomorphic vector field. For toric Fano varieties, we compute the Ding-Futaki invariants of toric degenerations, introduce relative Ding stability in a formal way. In the stable case, the Mabuchi metrics exist. Read More

In this paper, we study self-expanding solutions to a large class of parabolic inverse curvature flows by homogeneous symmetric functions of principal curvatures in Euclidean spaces. These flows include the inverse mean curvature flow and many nonlinear flows in the literature. We first show that the only compact self-expanders to any of these flows are round spheres. Read More

In this paper, we prove pointwise convergence of heat kernels for mGH-convergent sequences of $RCD^*(K,N)$-spaces. We obtain as a corollary results on the short-time behavior of the heat kernel in $RCD^*(K,N)$-spaces. We use then these results to initiate the study of Weyl's law in the $RCD$ setting Read More

In this paper, we study locally strongly convex centroaffine hypersurfaces with parallel cubic form with respect to the Levi-Civita connection of the centroaffine metric. As the main result, we obtain a complete classification of such centroaffine hypersurfaces. The result of this paper is a centroaffine version of the complete classification of locally strongly convex equiaffine hypersurfaces with parallel cubic form due to Hu, Li and Vrancken [12]. Read More

In this article, we prove that every compact simple Lie group $SO(n)$ for $n\geq 10$ admits at least $2\left([\frac{n-1}{3}]-2\right)$ non-naturally reductive left-invariant Einstein metrics. Read More

We survey different classification results for surfaces with parallel mean curvature immersed into some Riemannian homogeneous four-manifolds, including real and complex space forms, and product spaces. We provide a common framework for this problem, with special attention to the existence of holomorphic quadratic differentials on such surfaces. The case of spheres with parallel mean curvature is also explained in detail, as well as the state-of-the-art advances in the general problem. Read More

Generalising the results in arXiv:1612.00281, we construct infinite-dimensional families of non-singular stationary space times, solutions of Yang-Mills-Higgs-Einstein-Maxwell-Chern-Simons-dilaton-scalar field equations with a negative cosmological constant. The families include an infinite-dimensional family of solutions with the usual AdS conformal structure at conformal infinity. Read More

Lie-Butcher (LB) series are formal power series expressed in terms of trees and forests. On the geometric side LB-series generalizes classical B-series from Euclidean spaces to Lie groups and homogeneous manifolds. On the algebraic side, B-series are based on pre-Lie algebras and the Butcher-Connes-Kreimer Hopf algebra. Read More

For a given quasi-Fuchsian representation $\rho:\pi_1(S)\to$ PSL$_2\mathbb{C}$ of the fundamental group of a surface $S$ of genus $g\geq 2$, we prove that a generic branched complex projective structure on $S$ with holonomy $\rho$ and two branch points is obtained by bubbling some unbranched structure on $S$ with the same holonomy. Read More

Let $M$ be a compact orientable surface equipped with a volume form $\omega$, $P$ be either $\mathbb{R}$ or $S^1$, $f:M\to P$ be a $C^{\infty}$ Morse map, and $H$ be the Hamiltonian vector field of $f$ with respect to $\omega$. Let also $\mathcal{Z}_{\omega}(f) \subset C^{\infty}(M,\mathbb{R})$ be set of all functions taking constant values along orbits of $H$, and $\mathcal{S}_{\mathrm{id}}(f,\omega)$ be the identity path component of the group of diffeomorphisms of $M$ mutually preserving $\omega$ and $f$. We construct a canonical map $\varphi: \mathcal{Z}_{\omega}(f) \to \mathcal{S}_{\mathrm{id}}(f,\omega)$ being a homeomorphism whenever $f$ has at least one saddle point, and an infinite cyclic covering otherwise. Read More

The idea is to demonstrate the beauty and power of Alexandrov geometry by reaching interesting applications with a minimum of preparation. The applications include 1. Estimates on the number of collisions in billiards. Read More

In this short note, we prove that if $F$ is a weak upper semicontinuous admissible Finsler structure on a domain in $\mathbb{R}^n$, $n\geq 2$, then the intrinsic distance and differential structures coincide. Read More