We show that the zeroth coefficient of the cables of the HOMFLY polynomial (colored HOMFLY polynomials) does not distinguish mutants. This makes a sharp contrast with the total HOMFLY polynomial whose 3-cables can distinguish mutants.

In this paper, we consider the gradient estimates for the positive solutions to the following porous medium equation

$u_t = \Delta_V u^m$,

where $m>1$. We obtain Li-Yau type bounds of the above equation on Riemannian manifolds with Bakry-Emery type curvature bounded from below, which improves the estimates in [25] and covers the ones in [22, 18, 19, 27].

We establish the concept of $r$-almost Newton-Ricci soliton immersed into a Lorentzian manifold, which extends in a natural way the almost Ricci solitons introduced by Pigola, Rigoli, Rimoldi and Setti in [17]. In this setting, under suitable hypothesis on the potential and soliton functions, we obtain nonexistence and rigidity results. Some interesting examples of these new geometric objects are also given.

We prove that a projective vertical exact log smooth morphism of fs log analytic spaces with a base of log rank one yields polarized log Hodge structures in the canonical way.

This is the first of two articles in which we define an elliptically degenerating family of hyperbolic Riemann surfaces and study the asymptotic behavior of the associated spectral theory. Our study is motivated by a result which Hejhal attributes to Selberg, proving spectral accumulation for the family of Hecke triangle groups. In this article, we prove various results regarding the asymptotic behavior of heat kernels and traces of heat kernels for both real and complex time. In Garbin et al. (2018) [8], we will use the results from this article and study the asymptotic behavior of numerous spectral functions through elliptic degeneration, including spectral counting functions, Selberg's zeta function, Hurwitz-type zeta functions, determinants of the Laplacian, wave kernels, spectral projections, small eigenfunctions, and small eigenvalues. The method of proof we employ follows the template set in previous articles which study spectral theory on degenerating families of finite volume Riemann surfaces (Huntley et al. (1995) [14] and (1997) [15], Jorgenson et al. (1997) [20] and (1997) [17]) and on degenerating families of finite volume hyperbolic three manifolds (Dodziuk et al. (1998) [4].) Although the types of results developed here and in Garbin et al. (2018) [8], are similar to those in existing articles, it is necessary to thoroughly present all details in the setting of elliptic degeneration in order to uncover all nuances in this setting.

By a Pythagorean quadruplet $(a,b,c,d)$, we mean an integer solution to the quadratic equation $a^2 + b^2 = c^2 + d^2$. We use this notion to construct infinite families of elliptic curves of higher rank as far as possible. Furthermore, we give particular examples of rank eight.

Given a compact Riemannian manifold without boundary, in this paper, we discuss the monotonicity of the first eigenvalue of the $p$-Laplace operator under the Ricci-Bourguignon flow. We prove that the first eigenvalue of the $p$-Laplace operator is strictly monotone increasing and differentiable almost everywhere along the Ricci-Bourguignon flow under some different curvature assumptions. Moreover, we obtain various monotonicity quantities about the first eigenvalue of the $p$-Laplace operator along the Ricci-Bourguignon flow.

In this paper, we consider conditions under which a holomorphic motion of a closed subset of $\mathbf{Ĉ}$ over a non-simply connected Riemann surface $X$ can be extended to a holomorphic motion of $\mathbf{Ĉ}$ over $X$. We construct examples of non-extendable holomorphic motions which satisfy fairy good topological conditions. The examples are also counter-examples to a claim by Chirka for the extendability of holomorphic motions.

We focus on Lagrangian submanifolds of a six-dimensional sphere in the space Im $\mathbf {O}$ of imaginary octonions and study the relationship of such submanifolds with the geometry of the associative Grassmann manifold $\widetilde{{\rm Gr}}_{ass}$(Im $\mathbf {O}$) which is the Grassmann manifold of associative subspaces in Im $\mathbf {O}$. Considering the Gauss maps into $\widetilde{{\rm Gr}}_{ass}$(Im $\mathbf {O}$) associated to Lagrangian submanifolds, we show that those maps are harmonic. Moreover, the Gauss maps associated to homogeneous Lagrangian submanifolds are investigated.

We characterize a space-like surface in a pseudo-Riemannian space form with zero mean curvature vector, in terms of complex quadratic differentials on the surface as sections of a holomorphic line bundle. In addition, combining them, we have a holomorphic quartic differential. If the ambient space is $S^4$, then this differential is just one given in [5]. If the space is $S^4_1$, then the differential coincides with a holomorphic quartic differential in [6] on a Willmore surface in $S^3$ corresponding to the original surface through the conformal Gauss map. We define the conformal Gauss maps of surfaces in $E^3$ and $H^3$, and space-like surfaces in $S^3_1$, $E^3_1$, $H^3_1$ and the cone of future-directed light-like vectors of $E^4_1$, and have results which are analogous to those for the conformal Gauss map of a surface in $S^3$.