In this paper, we study some algebraic topology aspects of Stringc structures, more precisely, from the perspective of Whitehead tower and the perspective of the loop group of Spinc(n). We also extend the generalized Witten genera constructed for the first time by Chen et al. [J. Differential Geom. 88 (2011), pp. 1–40] to correspond to Stringc structures of various levels and give vanishing results for them.

Motivated by results on the simplicial volume of locally symmetric spaces of finite volume, in this note, we observe that the simplicial volume of the moduli space Mg,n is equal to 0 if g  2; g = 1, n  3; or g = 0, n  6; and the orbifold simplicial volume of Mg,n is positive if g = 1, n = 0, 1; g = 0, n = 4. We also observe that the simplicial volume of the Deligne-Mumford compactification of Mg,n is equal to 0, and the simplicial volumes of the reductive Borel-Serre compactification of arithmetic locally symmetric spaces ..\X and the Baily-Borel compactification of Hermitian arithmetic locally symmetric spaces ..\X are also equal to 0 if the Q-rank of ..\X is at least 3 or if ..\X is irreducible and of Q-rank 2.

In this paper, we prove that, a compact complex manifold $X$ admits a smooth Hermitian metric with positive (resp., negative) scalar curvature if and only if $K_X$ (resp., $K_X^{-1}$) is not pseudo-effective. On the contrary, we also show that on an arbitrary compact complex manifold $X$ with complex dimension $\geq 2$, there exist smooth Hermitian metrics with positive total scalar curvature, and one of the key ingredients in the proof relies on a recent solution to the Gauduchon conjecture by G. Székelyhidi, V. Tosatti, and B. Weinkove.

For all open Riemann surface N and real number  2 (0, /2), we construct a conformal minimal immersion X = (X1,X2,X3) : N ! R3 such that X3+tan()|X1| : N ! R is positive and proper. Furthermore, X can be chosen with an arbitrarily prescribed flux map. Moreover, we produce properly immersed hyperbolic minimal surfaces with non-empty boundary in R3 lying above a negative sublinear graph.

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