Volume 2014, Issue 692

Abstract. Soit L une extension finie non ramifiée de ℚ p . On s'intéresse au problème de savoir si certaines des représentations de GL2(L)$\operatorname{GL}_2(L)$ sur 𝔽p¯$\overline{\mathbb {F}_{p}}$ associées par Breuil et Paškūnas à une représentation de Gal(ℚp¯/L)${\operatorname{Gal}}(\overline{\mathbb {Q}_{p}}/L)$ de dimension 2 sur 𝔽p¯$\overline{\mathbb {F}_{p}}$ générique peuvent apparaître dans des espaces de formes automorphes.

Abstract. We define a new elliptic genus ψ on the complex bordism ring. With coefficients in ℤ[1/2]$\mathbb {Z}[1/2]$, we prove that it induces an isomorphism of the complex bordism ring modulo the ideal which is generated by all differences ℙ(E)-ℙ(E*)$\mathbb {P}(E)-\mathbb {P}(E^\ast )$ of projective bundles and their duals onto a polynomial ring on four generators in degrees 2, 4, 6 and 8. As an alternative geometric description of ψ, we prove that it is the universal genus which is multiplicative in projective bundles over Calabi–Yau 3-folds. With the help of the q -expansion of modular forms we will see that for a complex manifold M , the value ψ(M)$\psi (M)$ is a holomorphic Euler characteristic. We also compare ψ with Krichever–Höhn's complex elliptic genus and see that their only common specializations are Ochanine's elliptic genus and the χy$\chi _y$-genus.

Abstract. By the integral method we prove that any space-like entire graphic self-shrinking solution to the Lagrangian mean curvature flow in ℝn2n$\mathbb {R}^{2n}_{n}$ with the indefinite metric ∑idxidyi$\sum _i dx_idy_i$ is flat. This improves previous results by Huang–Wang [Calc. Var. Partial Differential Equations 41 (2011), 321–339] and Chau–Chen–Yuan [J. reine angew. Math. 664 (2012), 229–239] by removing the additional assumption in their results. In a similar manner, we reprove its Euclidean counterpart which is established by Chau–Chen–Yuan.

Abstract. Motivated by our subfactor generalization of Wall's conjecture and interest about completely rational conformal nets in general, in this paper we show that there is a one-to-one correspondence between intermediate conformal nets in a pair of completely rational nets 𝒜⊂ℬ$\mathcal {A}\subset \mathcal {B}$ with finite index and certain class of fusion rings. As a consequence we show that our conjecture for the pair 𝒜⊂ℬ$\mathcal {A}\subset \mathcal {B}$ is reduced to a conjecture about fusion rings, and our result also gives a method of enumerating all intermediate conformal nets.

Abstract. In this paper we prove a conjecture by Feldman–Ilmanen–Knopf (2003) that the gradient shrinking soliton metric they constructed on the tautological line bundle over ℂℙ1$\mathbb {CP}^1$ is the uniform limit of blow-ups of a type I Ricci flow singularity on a closed manifold. We use this result to show that limits of blow-ups of Ricci flow singularities on closed four-dimensional manifolds do not necessarily have non-negative Ricci curvature.

Abstract. A fundamental tool in the analysis of Ricci flow is a compactness result of Hamilton in the spirit of the work of Cheeger, Gromov and others. Roughly speaking it allows one to take a sequence of Ricci flows with uniformly bounded curvature and uniformly controlled injectivity radius, and extract a subsequence that converges to a complete limiting Ricci flow. A widely quoted extension of this result allows the curvature to be bounded uniformly only in a local sense. However, in this note we give a counterexample.

Abstract. We formally introduce the concept of localizing the Elliott conjecture at a given strongly self-absorbing C * -algebra 𝒟${\mathcal {D}}$; we also explain how the known classification theorems for nuclear C * -algebras fit into this concept. As a new result in this direction, we employ recent results of H. Lin to show that (under a mild K -theoretic condition) the class of separable, unital, simple C * -algebras with locally finite decomposition rank and UCT, and for which projections separate traces, satisfies the Elliott conjecture localized at the Jiang–Su algebra 𝒵$\mathcal {Z}$. Our main result is formulated in a more general way; this allows us to outline a strategy to possibly remove the trace space condition as well as the K -theory restriction entirely. When regarding both our result and the recent classification theorem of Elliott, Gong and Li as generalizations of the real rank zero case, the two approaches are perpendicular in a certain sense. The strategy to attack the general case aims at combining these two approaches. Our classification theorem covers simple ASH algebras for which projections separate traces (and the K -groups of which have finitely generated torsion part); it does, however, not at all depend on an inductive limit structure. Also, in the monotracial case it does not rely on the existence or absence of projections in any way. In fact, it is the first such result which, in a natural way, covers all known unital, separable, simple, nuclear and stably finite C * -algebras of real rank zero (the K -groups of which have finitely generated torsion part) as well as the (projectionless) Jiang–Su algebra itself.

Abstract. As an appendix to http://dx.doi.org/10.1515/crelle-2012-0082, this note provides some technical support to the proof of a result of W. Winter. It shows that two unital separable simple amenable 𝒵$\mathcal {Z}$-absorbing C * -algebras with locally finite decomposition property satisfying the UCT whose projections separate the traces are isomorphic if their K -theory is finitely generated and their Elliott invariants are the same.