International Journal of Mathematics

To an equisingularity class of complex plane branches, described by its multiplicity $n$ and characteristic exponents ${\beta }_1<\cdots <{\beta }_r$, $0<n<{\beta }_1$, there is a naturally associated family $K(n,{\beta }_1,\dots ,{\beta }_r)$ of equations containing a complete set of analytic representatives for all branches of the class. We show in this paper that the general polar curve of any member of $K(n,{\beta }_1,\dots ,{\beta }_r)$ is Newton degenerate, except when $r=1$, in which case the general member of $K(n,{\beta }_1)$ corresponds to a curve which has a Newton non-degenerate general polar curve with a fixed Newton polygon, or when $r=2$, $n=2p$, ${\beta }_1=2q$, ${\beta }_2=2q+d$, with $\mathrm{\text{GCD}}(p,q)=1$ and $d\ge 1$ is odd, in which case $K(2p,2q,2q+d)$ has a subset containing a complete set of analytic representatives for all branches of the class whose general member has also a Newton non-degenerate general polar curve with a fixed Newton polygon. In both cases, we give explicit Zariski open sets the points of which represent branches with Newton non-degenerate polars and describe the topology of their general polars.

We provide classification results for and examples of half conformally flat generalized quasi-Einstein manifolds of signature $(2,2)$. This analysis leads to a natural equation in affine geometry called the affine quasi-Einstein equation that we explore in further detail.

In this paper, a new notion of isotropic curvature for sprays is introduced. We show that for a spray of scalar curvature, it is of isotropic curvature if and only if the non-Riemannian quantity $\chi$ vanishes. In fact, it is the first geometric quantity to show the spray of isotropic curvature even in the Finslerian case. How to determine a spray is induced by a Finsler metric or not is an interesting inverse problem. We study this problem when the spray is of isotropic curvature and show that a spray of zero curvature can be induced by a group of Finsler metrics. Further, an efficient way is given to construct a family of sprays of isotropic curvature which cannot be induced by any Finsler metric.

We show that the Friedrichs operator exhibits smoothing properties in the $L^p$ scale. In particular, we prove that on any smoothly bounded pseudoconvex domain the Friedrichs operator maps $A^2(\mathrm{\Omega })$ to $A^p(\mathrm{\Omega })$ for some $p>2$.

This paper presents a systematic study for abstract Banach measure algebras over homogeneous spaces of compact groups. Let $H$ be a closed subgroup of a compact group $G$ and $G/H$ be the left coset space associated to the subgroup $H$ in $G$. Also, let $M(G/H)$ be the Banach measure space consists of all complex measures over $G/H$. Then we introduce the abstract notions of convolution and involution over the Banach measure space $M(G/H)$.

Let $(M,\overline{g})$ be a Kähler surface, and $\mathrm{\Sigma }$ an immersed surface in $M$. The Kähler angle of $\mathrm{\Sigma }$ in $M$ is introduced by Chern and Wolfson [Am. J. Math. 105 (1983) 59–83]. Let $(M,\overline{g}(t))$ evolve along the Kähler–Ricci flow, and ${\mathrm{\Sigma }}_t$ in $(M,\overline{g}(t))$ evolve along the mean-curvature flow. We show that the Kähler angle $\alpha (t)$ satisfies the evolution equation

The equation implies that if the initial surface is symplectic (Lagrangian), then, along the flow, ${\mathrm{\Sigma }}_t$ is always symplectic (Lagrangian) at each time $t$, which we call a symplectic (Lagrangian) Kähler–Ricci mean-curvature flow.

In this paper, we mainly study the symplectic Kähler–Ricci mean-curvature flow.

A framed symplectic sheaf on a smooth projective surface $X$ is a torsion-free sheaf $E$ together with a trivialization on a divisor $D\subseteq X$ and a morphism ${\mathrm{\Lambda }}^{2}E\to {{𝒪}}_X$ satisfying some additional conditions. We construct a moduli space for framed symplectic sheaves on a surface, and present a detailed study for $X={\mathbb{P}}_{ℂ}^2$. In this case, the moduli space is irreducible and admits an ADHM-type description and a birational proper map onto the space of framed symplectic ideal instantons.

We prove an upper bound for geodesic periods of Maass forms over hyperbolic manifolds. By definition, such periods are integrals of Maass forms restricted to a special geodesic cycle of the ambient manifold, against a Maass form on the cycle. Under certain restrictions, the bound will be uniform.