# Differential Geometry

## New submissions

[ total of 29 entries: 1-29 ]
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### New submissions for Tue, 20 Mar 18

[1]
Title: Geometry of probability simplex via optimal transport
Authors: Wuchen Li
Subjects: Differential Geometry (math.DG)

We study the Riemannian structures of the probability simplex on a weighted graph introduced by $L^2$-Wasserstein metric. The main idea is to embed the probability simplex as a submanifold of the positive orthant. From this embedding, we establish the geometry formulas of the probability simplex in Euclidean coordinates. The geometry computations on discrete simplex guide us to introduce the ones in the Fr{\'e}chet manifold of densities supported on a finite dimensional base manifold. Following the steps of Nelson, Bakery-{\'E}mery, Lott-Villani-Strum and the geometry of density manifold, we demonstrate an identity that connects the Bakery-{\'E}mery $\Gamma_2$ operator (carr{\'e} du champ it{\'e}r{\'e}) and Yano's formula on the base manifold. Several examples of differential equations in probability simplex are demonstrated.

[2]
Title: Singular genuine rigidity
Subjects: Differential Geometry (math.DG)

We extend the concept of genuine rigidity of submanifolds by allowing mild singularities, mainly to obtain new global rigidity results and unify the known ones. As one of the consequences, we simultaneously extend and unify Sacksteder and Dajczer-Gromoll theorems by showing that any compact $n$-dimensional submanifold of ${\mathbb R}^{n+p}$ is singularly genuinely rigid in ${\mathbb R}^{n+q}$, for any $q < \min\{5,n\} - p$. Unexpectedly, the singular theory becomes much simpler and natural than the regular one, even though all technical codimension assumptions, needed in the regular case, are removed.

[3]
Title: Connection Blocking In Quotients of $Sol$
Comments: 10 pages. arXiv admin note: text overlap with arXiv:1706.07996; text overlap with arXiv:1211.7291 by other authors
Subjects: Differential Geometry (math.DG)

Let $G$ be a connected Lie group and $\Gamma \subset G$ a lattice. Connection curves of the homogeneous space $M=G/\Gamma$ are the orbits of one parameter subgroups of $G$. To $block$ a pair of points $m_1,m_2 \in M$ is to find a finite set $B \subset M\setminus \{m_1, m_2 \}$ such that every connecting curve joining $m_1$ and $m_2$ intersects $B$. The homogeneous space $M$ is $blockable$ if every pair of points in $M$ can be blocked, otherwise we call it $non-blockable$. $Sol$ is an important Lie group and one of the eight homogeneous Thurston 3-geometries. It is a unimodular solvable Lie group diffeomorphic to $R^3$, and together with the left invariant metric $ds^2=e^{-2z}dx^2+e^{2z}dy^2+dz^2$ includes copies of the hyperbolic plane, which makes studying its geometrical properties more interesting. In this paper we prove that all quotients of $Sol$ are non-blockable. In particular, we show that for any lattice $\Gamma \subset Sol$, the set of non-blockable pairs is a dense subset of $Sol/\Gamma \times Sol/\Gamma$.

[4]
Title: Transversely holomorphic branched Cartan geometry
Subjects: Differential Geometry (math.DG); Complex Variables (math.CV)

Earlier we introduced and studied the concept of holomorphic {\it branched Cartan geometry}. We define here a foliated version of this notion; this is done in terms of Atiyah bundle. We show that any complex compact manifold of algebraic dimension $d$ admits, away from a closed analytic subset of positive codimension, a nonsingular holomorphic foliation of complex codimension $d$ endowed with a transversely flat branched complex projective geometry (equivalently, a ${\mathbb C}P^d$-geometry). We also prove that transversely branched holomorphic Cartan geometries on compact complex projective rationally connected varieties and on compact simply connected Calabi-Yau manifolds are always flat (consequently, they are defined by holomorphic maps into homogeneous spaces).

[5]
Title: Toric geometry of $G_2$-manifolds
Subjects: Differential Geometry (math.DG); High Energy Physics - Theory (hep-th)

We consider $G_2$-manifolds with an effective torus action that is multi-Hamiltonian for one or more of the defining forms. The case of $T^3$-actions is found to be distinguished. For such actions multi-Hamiltonian with respect to both the three- and four-form, we derive a Gibbons-Hawking type ansatz giving the geometry on an open dense set in terms a symmetric $3\times 3$-matrix of functions. This leads to particularly simple examples of explicit metrics with holonomy equal to $G_2$. We prove that the multi-moment maps exhibit the full orbit space topologically as a smooth four-manifold containing a trivalent graph as the image of the set of special orbits and describe these graphs in some complete examples.

[6]
Title: Conformal slant submersions in contact geometry
Subjects: Differential Geometry (math.DG)

Akyol M.A. [Conformal anti-invariant submersions from cosymplectic manifolds, Hacettepe Journal of Mathematics and Statistic, 46(2), (2017), 177-192.] defined and studied conformal anti-invariant submersions from cosymplectic manifolds. The aim of the present paper is to define and study the notion of conformal slant submersions (it means the Reeb vector field $\xi$ is a vertical vector field) from almost contact metric manifolds onto Riemannian manifolds as a generalization of Riemannian submersions, horizontally conformal submersions, slant submersions and conformal anti-invariant submersions. More precisely, we mention lots of examples and obtain the geometries of the leaves of $\ker\pi_{*}$ and $(\ker\pi_{*})^\perp,$ including the integrability of the distributions, the geometry of foliations, some conditions related to totally geodesicness and harmonicty of the submersions. Finally, we consider a decomposition theorem on total space of the new submersion.

[7]
Title: Shifted Poisson structures on differentiable stacks
Subjects: Differential Geometry (math.DG)

The purpose of this paper is to investigate shifted $(+1)$ Poisson structures in context of differential geometry. The relevant notion is shifted $(+1)$ Poisson structures on differentiable stacks. More precisely, we develop the notion of Morita equivalence of quasi-Poisson groupoids. Thus isomorphism classes of $(+1)$ Poisson stack correspond to Morita equivalence classes of quasi-Poisson groupoids. In the process, we carry out the following programs of independent interests:
(1) We introduce a $\mathbb Z$-graded Lie 2-algebra of polyvector fields on a given Lie groupoid and prove that its homotopy equivalence class is invariant under Morita equivalence of Lie groupoids, thus can be considered as polyvector fields on the corresponding differentiable stack ${\mathfrak X}$. It turns out that shifted $(+1)$ Poisson structures on ${\mathfrak X}$ correspond exactly to elements of the Maurer-Cartan moduli set of the corresponding dgla.
(2) We introduce the notion of tangent complex $T_{\mathfrak X}$ and cotangent complex $L_{\mathfrak X}$ of a differentiable stack ${\mathfrak X}$ in terms of any Lie groupoid $\Gamma{\rightrightarrows} M$ representing ${\mathfrak X}$. They correspond to homotopy class of 2-term homotopy $\Gamma$-modules $A[1]\rightarrow TM$ and $T^\vee M\rightarrow A^\vee[-1]$, respectively. We prove that a $(+1)$-shifted Poisson structure on a differentiable stack ${\mathfrak X}$, defines a morphism ${L_{{\mathfrak X}}}[1]\to {T_{{\mathfrak X}}}$. We rely on the tools of theory of VB-groupoids including homotopy and Morita equivalence of VB-groupoids.

[8]
Title: Height estimates for mean curvature graphs in $\mathrm{Nil}_3$ and $\widetilde{PSL}_2(\mathbb{R})$
Authors: Antonio Bueno
Subjects: Differential Geometry (math.DG)

In this paper we obtain height estimates for compact, constant mean curvature vertical graphs in the homogeneous spaces $\mathrm{Nil}_3$ and $\widetilde{PSL}_2(\mathbb{R})$. As a straightforward consequence, we announce a structure-type result for proper graphs defined on relatively compact domains.

[9]
Title: Higher-order estimates for collapsing Calabi-Yau metrics
Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP); Complex Variables (math.CV)

We prove a uniform C^alpha estimate for collapsing Calabi-Yau metrics on the total space of a proper holomorphic submersion over the unit ball in C^m. The usual methods of Calabi, Evans-Krylov, and Caffarelli do not apply to this setting because the background geometry degenerates. We instead rely on blowup arguments and on linear and nonlinear Liouville theorems on cylinders. In particular, as an intermediate step, we use such arguments to prove sharp new Schauder estimates for the Laplacian on cylinders. If the fibers of the submersion are pairwise biholomorphic, our method yields a uniform C^infinity estimate. We then apply these local results to the case of collapsing Calabi-Yau metrics on compact Calabi-Yau manifolds. In this global setting, the C^0 estimate required as a hypothesis in our new local C^alpha and C^infinity estimates is known to hold thanks to earlier work of the second-named author.

[10]
Title: On the existence problem of Einstein-Maxwell Kähler metrics
Subjects: Differential Geometry (math.DG)

In this expository paper we review on the existence problem of Einstein-Maxwell K\"ahler metrics, and make several remarks. Firstly, we consider a slightly more general set-up than Einstein-Maxwell K\"ahler metrics, and give extensions of volume minimization principle, the notion of toric K-stability and other related results to the general set-up. Secondly, we consider the toric case when the manifold is the one point blow-up of the complex project plane and the K\"ahler class $\Omega$ is chosen so that the area of the exceptional curve is sufficiently close to the area of the rational curve of self-intersection number 1. We observe by numerical analysis that there should be a Killing vector field $K$ which gives a toric K-stable pair $(\Omega, K)$ in the sense of Apostolov-Maschler.

[11]
Title: Higher symmetries of symplectic Dirac operator
Comments: Symplectic Dirac operator, Higher symmetry algebra, Projective differential geometry, Minimal nilpotent orbit, $\mathfrak{sl}(3,\mR)$
Subjects: Differential Geometry (math.DG); Mathematical Physics (math-ph); Functional Analysis (math.FA); Representation Theory (math.RT); Symplectic Geometry (math.SG)

We construct in projective differential geometry of the real dimension $2$ higher symmetry algebra of the symplectic Dirac operator ${D}\kern-0.5em\raise0.22ex\hbox{/}_s$ acting on symplectic spinors. The higher symmetry differential operators correspond to the solution space of a class of projectively invariant overdetermined operators of arbitrarily high order acting on symmetric tensors. The higher symmetry algebra structure corresponds to a completely prime primitive ideal having as its associated variety the minimal nilpotent orbit of $\mathfrak{sl}(3,{\mathbb{R}})$.

[12]
Title: Maximal Symmetry and Unimodular Solvmanifolds
Subjects: Differential Geometry (math.DG)

Recently, it was shown that Einstein solvmanifolds have maximal symmetry in the sense that their isometry groups contain the isometry groups of any other left-invariant metric on the given Lie group. Such a solvable Lie group is necessarily non-unimodular. In this work we consider unimodular solvable Lie groups and prove that there is always some metric with maximal symmetry. Further, if the group at hand admits a Ricci soliton, then it is the isometry group of the Ricci soliton which is maximal.

### Cross-lists for Tue, 20 Mar 18

[13]  arXiv:1803.06382 (cross-list from math.GT) [pdf, ps, other]
Title: Harmonic spinors on the Davis hyperbolic 4-manifold
Comments: 33 pages and 2 figures
Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

In this paper we use the G-spin theorem to show that the Davis hyperbolic 4-manifold admits harmonic spinors. This is the first example of a closed hyperbolic 4-manifold that admits harmonic spinors. We also explicitly describe the Spinor bundle of a spin hyperbolic 2- or 4-manifold and show how to calculated the subtle sign terms in the G-spin theorem for an isometry, with isolated fixed points, of a closed spin hyperbolic 2- or 4-manifold.

[14]  arXiv:1803.06499 (cross-list from math.CV) [pdf, ps, other]
Title: Kähler submanifolds of the symmetrized polydisc
Comments: To appear in Comptes Rendus Mathematique
Subjects: Complex Variables (math.CV); Differential Geometry (math.DG)

This paper proves the non-existence of common K\"ahler submanifolds of the complex Euclidean space and the symmetrized polydisc endowed with their canonical metrics.

[15]  arXiv:1803.06556 (cross-list from math.CA) [pdf, ps, other]
Title: Linearization of third-order ordinary differential equations u'''=f(x,u,u',u'') via point transformations
Subjects: Classical Analysis and ODEs (math.CA); Differential Geometry (math.DG)

The linearization problem by use of the Cartan equivalence method for scalar third-order ODEs via point transformations was solved partially in [1,2]. In order to solve this problem completely, the Cartan equivalence method is applied to provide an invariant characterization of the linearizable third-order ordinary differential equation u'''=f(x,u,u',u'') which admits a four-dimensional point symmetry Lie algebra. The invariant characterization is given in terms of the function f in a compact form. A simple procedure to construct the equivalent canonical form by use of an obtained invariant is also presented. The method provides auxiliary functions which can be utilized to efficiently determine the point transformation that does the reduction to the equivalent canonical form. Furthermore, illustrations to the main theorem and applications are given.

[16]  arXiv:1803.06582 (cross-list from math.MG) [pdf, other]
Title: Contrasting Various Notions of Convergence in Geometric Analysis
Comments: 7 figures by Penelope Chang of Hunter College High School
Subjects: Metric Geometry (math.MG); Differential Geometry (math.DG)

We explore the distinctions between $L^p$ convergence of metric tensors on a fixed Riemannian manifold versus Gromov-Hausdorff, uniform, and intrinsic flat convergence of the corresponding sequence of metric spaces. We provide a number of examples which demonstrate these notions of convergence do not agree even for two dimensional warped product manifolds with warping functions converging in the $L^p$ sense. We then prove a theorem which requires $L^p$ bounds from above and $C^0$ bounds from below on the warping functions to obtain enough control for all these limits to agree.

[17]  arXiv:1803.06866 (cross-list from math-ph) [pdf, ps, other]
Title: The planar 3-body problem II:reduction to pure shape and spherical geometry (2nd version)
Subjects: Mathematical Physics (math-ph); Differential Geometry (math.DG)

Geometric reduction of the Newtonian planar three-body problem is investigated in the framework of equivariant Riemannian geometry, which reduces the study of trajectories of three-body motions to the study of their moduli curves, that is, curves which record the change of size and shape, in the moduli space of oriented mass-triangles. The latter space is a Riemannian cone over the shape 2-sphere, and the shape curve is the image curve on this sphere. It is shown that the time parametrized moduli curve is in general determined by the relative geometry of the shape curve and the shape potential function. This also entails the reconstruction of time, namely the geometric shape curve determines the time parametrization of the moduli curve, hence also the three-body motion itself, modulo a fixed rotation of the plane. The first version of this work is an (unpublished) paper from 2012, and the present version is an editorial revision of this.

### Replacements for Tue, 20 Mar 18

[18]  arXiv:1408.1989 (replaced) [pdf, ps, other]
Title: On the stability of L^p-norms of Curvature Tensor at Rank one symmetrics spaces
Authors: Soma Maity
Comments: 15 Pages and 1 figure
Subjects: Differential Geometry (math.DG)
[19]  arXiv:1610.00351 (replaced) [pdf, ps, other]
Title: Quantitative stratification of stationary connections
Authors: Yu Wang
Subjects: Differential Geometry (math.DG)
[20]  arXiv:1708.07573 (replaced) [pdf, other]
Title: Reconstruction of a compact Riemannian manifold from the scattering data of internal sources
Subjects: Differential Geometry (math.DG)
[21]  arXiv:1710.05318 (replaced) [pdf, ps, other]
Title: On Finsler spacetimes with a timelike Killing vector field
Comments: 28 pages, AMSLaTex. v3: First part of the introduction and conclusions section expanded, some new references added; v3 matches the published version
Journal-ref: Class. Quantum Grav. 35 (2018) 085007 (28pp)
Subjects: Differential Geometry (math.DG); General Relativity and Quantum Cosmology (gr-qc)
[22]  arXiv:1710.05537 (replaced) [pdf, ps, other]
Title: Hamiltonian stability for weighted measure and generalized Lagrangian mean curvature flow
Comments: 40 pages; (ver.3) minor corrections. (ver.2) Tex file format is changed, Corollary 2.5 and a reference are added, minor corrections
Subjects: Differential Geometry (math.DG)
[23]  arXiv:1711.08024 (replaced) [pdf, ps, other]
Title: New complex analytic methods in the theory of minimal surfaces: a survey
Comments: To appear in J. Aust. Math. Soc
Subjects: Differential Geometry (math.DG); Complex Variables (math.CV)
[24]  arXiv:1802.06848 (replaced) [pdf, ps, other]
Title: Stable constant mean curvature surfaces with free boundary in slabs
Authors: Rabah Souam
Subjects: Differential Geometry (math.DG)
[25]  arXiv:1802.08922 (replaced) [pdf, ps, other]
Title: Positively curved Killing foliations via deformations
Comments: 24 pages, corrected typos, example 8.1 removed
Subjects: Differential Geometry (math.DG)
[26]  arXiv:1802.10248 (replaced) [pdf, ps, other]
Title: M-eigenvalues of The Riemann Curvature Tensor
Subjects: Differential Geometry (math.DG); Spectral Theory (math.SP)
[27]  arXiv:1803.00856 (replaced) [pdf, ps, other]
Title: Embedded loops in the hyperbolic plane with prescribed, almost constant curvature
Comments: Theorems numbering has been changed; few misprints were fixed
Subjects: Differential Geometry (math.DG)
[28]  arXiv:1707.00977 (replaced) [pdf, ps, other]
Title: Electric-Magnetic Aspects On Yang-Mills Fields
Authors: Tosiaki Kori