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Register a new userManifolds for which Hubers Theorem holds
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Extensions of Hubers finite-point conformal compactification theorem to higher dimensions with $L^frac{n}{2}$ bounded scalar curvature have been studied for many years. In this paper, we discuss the properties of conformal metrics with $|R|_{L^frac{n}{2}}<+infty$ on a punctured ball of a Riemannian manifold to find some geometric obstacles for Hubers theorem. To our surprise, such metrics are rather more rigid than we have ever thought. For example, their volume densities at infinity are exact 1, which implies that Carron and Herzlichs Euclidean volume growth condition is also a necessary condition for Hubers Theorem. When the dimension is 4, we derive the $L^2$-integrability of Ricci curvature, which follows that the Pfaffian of the curvature is integrable and satisfies a Gauss-Bonnet-Chern formula. We also prove that the Gauss-Bonnet-Chern formula proved by Lu and Wang, under the assumption that the second fundamental form is in $L^4$, holds when $Rin L^2$.
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We introduce polar metrics on a product manifold, which have product and warped product metrics as special cases. We prove a de Rham-type theorem characterizing Riemannian manifolds that can be locally decomposed as a product manifold endowed with a polar metric. For a product manifold endowed with a polar metric, our main result gives a complete description of all its isometric immersions into a space form whose second fundamental forms are adapetd to its product structure, in the sense that the tangent spaces to each factor are preserved by all shape operators. This is a far-reaching generalization of a basic decomposition theorem for isometric immersions of Riemannian products due to Moore as well as its extension by Nolker to isometric immersions of warped products.
Mosers theorem (1965) states that the diffeomorphism group of a compact manifold acts transitively on the space of all smooth positive densities with fixed volume. Here we describe the extension of this result to manifolds with corners. In particular we obtain Mosers theorem on simplices. The proof is based on Banyagas paper (1974), where Mosers theorem is proven for manifolds with boundary. A cohomological interpretation of Banyagas operator is given, which allows a proof of Lefschetz duality using differential forms.
We study the index of the APS boundary value problem for a strongly Callias-type operator $D$ on a complete even dimensional Riemannian manifold $M$ (the odd dimensional case was considered in our previous paper arXiv:1706.06737). We use this index to define the relative $eta$-invariant $eta(A_1,A_0)$ of two strongly Callias-type operators, which are equal outside of a compact set. Even though in our situation the $eta$-invariants of $A_1$ and $A_0$ are not defined, the relative $eta$-invariant behaves as if it were the difference $eta(A_1)-eta(A_0)$. We also define the spectral flow of a family of such operators and use it compute the variation of the relative $eta$-invariant.
An analogue of the Stefan-Sussmann Theorem on manifolds with boundary is proven for normal distributions. These distributions contain vectors transverse to the boundary along its entirety. Plain integral manifolds are not enough to integrate a normal distribution; the next best integrals are so-called neat integral manifolds with boundary. The conditions on the distribution for this integrability is expressed in terms of adapted collars and integrability of a pulled-back distribution on the interior and on the boundary.
In this paper, we prove a Liouville theorem for holomorphic functions on a class of complete Gauduchon manifolds. This generalizes a result of Yau for complete Kahler manifolds to the complete non-Kahler case.
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