In this paper, we present two kinds of total Chern forms c(E,G) and c(E,G) as well as a total Segre form c(E,G) of a holomorphic Finsler vector bundle c(E,G) expressed by the Finsler metric c(E,G), which answers a question of Faran [The equivalence problem for complex Finsler Hamiltonians, in <i>Finsler Geometry</i>, Contemporary Mathematics, Vol. 196 (American Mathematical Society, Providence, RI, 1996), pp. 133144] to some extent. As some applications, we show that the signed Segre forms c(E,G) are positive c(E,G)-forms on c(E,G) when c(E,G) is of positive Kobayashi curvature; we prove, under an extra assumption, that a FinslerEinstein vector bundle in the sense of Kobayashi is semi-stable; we introduce a new definition of a flat Finsler metric, which is weaker than Aikous one [Finsler geometry on complex vector bundles, in <i>A Sampler of RiemannFinsler Geometry</i>, MSRI Publications, Vol. 50 (Cambridge

In this paper, we give a complete classification of cotorsion pairs in a cluster category C of type A^\infty_\infty via certain configurations of arcs, called τ-compact Ptolemy diagrams, in an infinite strip with marked points. As applications, we classify t-structures and functorially finite rigid subcategories in C, respectively. We also deduce Liu-Paquette's classification of cluster tilting subcategories of C and Ng's classification of torsion pairs in the cluster category of type A^\infty_\infty.

New spaces of Lipschitz type on metric-measure spaces are introduced and they are shown to be just the well-known Besov spaces or Triebel–Lizorkin spaces when the smooth index is less than 1. These theorems also hold in the setting of spaces of homogeneous type, which include Euclidean spaces, Riemannian manifolds and some self-similar fractals. Moreover, the relationships amongst these Lipschitz-type spaces, Hajłasz–Sobolev spaces, Korevaar–Schoen–Sobolev spaces, Newtonian Sobolev space and Cheeger–Sobolev spaces on metric-measure spaces are clarified, showing that they are the same space with equivalence of norms. Furthermore, a Sobolev embedding theorem, namely that the Lipschitz-type spaces with large orders of smoothness can be embedded in Lipschitz spaces, is proved. For metric-measure spaces with heat kernels, a Hardy–Littlewood–Sobolev theorem is establish, and hence it is proved that Lipschitz-type spaces with small orders of smoothness can be embedded in Lebesgue spaces.

On a complete noncompact K¨ahler manifold we prove that the bottom of the spectrum for the Laplacian is bounded from above by m2 if the Ricci curvature is bounded from below by −2(m+1). Then we show that if this upper bound is achieved then either the manifold is connected at infinity or it has two ends and in this case it is diffeomorphic to the product of the real line with a compact manifold and we determine the metric.

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