Recently Freed and Hopkins [11] proved that there is no parity anomaly in M-theory on pin+ manifolds in the low-energy field theory approximation, and they also developed an algebraic theory of cubic forms. Earlier Witten [33] proved the anomaly cancellation for spin manifolds by introducing the E8-bundle technique. Motivated by the cubic forms and the anomaly cancellation formulas of Witten-Freed-Hopkins, we give some new cubic forms on spin, spinc, spinω2 and orientable 12- manifolds respectively. We relate them to η-invariants when the manifolds are with boundary, and mod 2 indices on 10 dimensional characteristic submanifolds when the manifolds are spinc or spinω2 . Our method of producing these cubic forms is a combination of (generalized) Witten classes and the character of the basic representation of affine E8.

Existence and uniqueness of the solution to the Lp Minkowski problem for the electrostatic p-capacity are proved when p > 1 and 1 < p < n. These results are nonlinear extensions of the very recent solution to the Lp Minkowski problem for p-capacity when p = 1 and 1 < p < n by Colesanti et al. and Akman et al., and the classical solution to the Minkowski problem for electrostatic capacity when p = 1 and p = 2 by Jerison.

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The purpose of this paper is to classify the Möbius homogeneous hypersurfaces with two distinct principal curvatures in$S$^{$n$+1}under the Möbius transformation group. Additionally, we give a classification of the Möbius homogeneous hypersurfaces in$S$^{4}.

In the general, this article addresses the nature of, and relation between, canonical and log-canonical singularities of foliations by curves. The case of equivariance under a finite group action has several peculiarities, amongst which are examples from dimension 3 onward where no birational modification can be both Gorenstein and log-canonical. The article in the specific, i.e. a functorial resolution algorithm for foliated 3-folds in the 2-category of Deligne-Mumford champs, is thus not only optimal but natural.