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We prove that for every non-negative integer g, there exists a bound on the number of ends of a complete, embedded minimal surface M in R3 of genus g and finite topology. This bound on the finite number of ends when M has at least two ends implies that M has finite stability index which is bounded by a constant that only depends on its genus.

In this paper, we will establish a regularity theory for the Kähler–Ricci flow on Fano$n$-manifolds with Ricci curvature bounded in$L$^{$p$}-norm for some $${p > n}$$ . Using this regularity theory, we will also solve a long-standing conjecture for dimension 3. As an application, we give a new proof of the Yau–Tian–Donaldson conjecture for Fano 3-manifolds. The results have been announced in [45].

We establish a general Kronecker limit formula of arbitrary rank over global function fields with Drinfeld period domains playing the role of upper-half plane. The Drinfeld-Siegel units come up as equal characteristic modular forms replacing the classical $\Delta$. This leads to analytic means of deriving a Colmez-type formula for "stable Taguchi height" of CM Drinfeld modules having arbitrary rank. A Lerch-Type formula for "totally real" function fields is also obtained, with the Heegner cycle on the Bruhat-Tits buildings intervene. Also our limit formula is naturally applied to the special values of both the Rankin-Selberg L-functions and the Godement-Jacquet L-functions associated to automorphic cuspidal representations over global function fields.