Differential Geometry and its Applications最新文献

We study complex analytic projective connections on the plane. We characterize some of them in terms of their families of integral curves. We also give a beginning of classification of second order odes polynomial in the first and second derivatives, and with holomorphic coefficients.

In Euclidean geometry, the shortest distance between two points is a straight line. Chern made a conjecture (cf. [11]) in 1977 that an affine maximal graph of a smooth and locally uniformly convex function on two-dimensional Euclidean space $R^2$ must be a paraboloid. In 2000, Trudinger and Wang completed the proof of this conjecture in affine geometry (cf. [47]). (Caution: in these literatures, the term “affine geometry” refers to “equi-affine geometry”.) A natural problem arises: Whether the hyperbola is a general-affine maximal curve in $R^2$? In this paper, by utilizing the evolution equations for curves, we obtain the second variational formula for general-affine extremal curves in $R^2$, and show the general-affine maximal curves in $R^2$ are much more abundant and include the explicit curves $y=x^{\alpha }(\alpha \text{is a constant and}\alpha \notin \{0,1,\frac{1}{2},2\left\}\right)$ and $y=x\log x$. At the same time, we generalize the fundamental theory of curves in higher dimensions, equipped with $\text{GA}\left(n\right)=\text{GL}\left(n\right)⋉R^n$. Moreover, in general-affine plane geometry, an isoperimetric inequality is investigated, and a complete classification of the solitons for general-affine heat flow is provided. We also study the local existence, uniqueness, and long-term behavior of this general-affine heat flow. A closed embedded curve will converge to an ellipse when evolving according to the general-affine heat flow is proved.

A twistor lift of a space-like or time-like surface in a neutral hyperKähler 4-manifold with zero mean curvature vector is given by a (para)holomorphic function, which yields (para)holomorphicity of the Gauss maps of space-like or time-like surfaces in $E_2^4$ with zero mean curvature vector. For a space-like or time-like surface in an oriented neutral 4-manifold with zero mean curvature vector such that both twistor lifts belong to the kernel of the curvature tensor, its (para)complex quartic differential is holomorphic. If both twistor lifts of a time-like surface with zero mean curvature vector have light-like or zero covariant derivatives, then either the shape operator with respect to a light-like normal vector field vanishes or all the shape operators of the surface are light-like or zero. Examples with the former (resp. latter) property are given by the conformal Gauss maps of time-like surfaces of Willmore type with zero paraholomorphic quartic differential (resp. time-like surfaces in 4-dimensional neutral space forms based on the Gauss-Codazzi-Ricci equations).

We expect that every Cartan–Münzner polynomial of degree four can be described as a squared-norm of a moment map for a Hamiltonian action. Our expectation is known to be true for Hermitian cases, that is, those obtained from the isotropy representations of compact irreducible Hermitian symmetric spaces of rank two. In this paper, we prove that our expectation is true for the Cartan–Münzner polynomials obtained from the isotropy representations of Grassmannian manifolds of rank two over $R$, $C$ or $H$. The quaternion cases are the first non-Hermitian examples that our expectation is verified.

We study Rarita-Schwinger fields on 6-dimensional compact strict nearly Kähler manifolds. In order to investigate them, we clarify the relationship between some differential operators for the Hermitian connection and the Levi-Civita connection. As a result, we show that the space of Rarita-Schwinger fields coincides with the space of harmonic 3-forms. Applying the same technique to deformation theory, we also find that the space of infinitesimal deformations of Killing spinors coincides with the direct sum of a certain eigenspace of the Laplace operator and the space of Killing spinors.

We express Darboux transformations of discrete polarised curves as parallel sections of discrete connections in the quaternionic formalism. This immediately leads to the linearisation of the monodromy of the transformation. We also consider the integrable reduction to the case of discrete bicycle correspondence. Applying our method to the case of discrete circles, we obtain closed-form discrete parametrisations of all (closed) Darboux transforms and (closed) bicycle correspondences.

We present new expressions for the integrals of mean curvature of domains in $R^n$ by means of sections with cylinders. Then, we combine these expressions with the corresponding version of the invariant density of affine subspaces in $R^n$, in order to obtain pseudo-rotational formulae for all the integrals of mean curvature of ∂K. As particular cases, we present pseudo-rotational integral formulas for the volume, area, integral of mean curvature, and Euler-Poincaré characteristic of a connected domain of $R^3$, whose boundary is a surface, considering slabs in $R^3$ whose central plane passes through a fixed point, and cylinders contained in these slabs.

We estimate explicit lower bounds for the isoperimetric profiles of the Riemannian product of a compact manifold and the Euclidean space with the flat metric, $(M^m\times R^n,g+g_E)$, $m,n>1$. In particular, we introduce a lower bound for the isoperimetric profile of $M^m\times R^n$ for regions of large volume and we improve on previous estimates of lower bounds for the isoperimetric profiles of $S^2\times R^2$, $S^3\times R^2$, $S^2\times R^3$. We also discuss some applications of these results in order to improve known lower bounds for the Yamabe invariant of certain product manifolds.

It is known that Mendonça and Tojeiro (2013) [19] have established a complete classification of parallel submanifolds in the product manifold $Q_{k_1}^{n_1}\times Q_{k_2}^{n_2}$, where $Q_{k_1}^{n_1}$ (resp. $Q_{k_2}^{n_2}$) is an $n_1$-dimensional (resp. $n_2$-dimensional) real space form with constant curvature $k_1$ (resp. $k_2$). In this paper, motivated by this result with considering further generalization, we study those semi-parallel hypersurfaces in case $Q_{k_1}^{n_1}=S_{k_1}^{n_1}$ and $Q_{k_2}^{n_2}=S_{k_2}^{n_2}$ with $k_1,k_2>0$. As the main result, we classify semi-parallel hypersurfaces of