An analytical proof of Kodaira's embedding theorem for Hodge manifolds

A proof of Sobolev’s Embedding Theorem for Compact Riemannian Manifolds

Observe that H 0 (M) = L p(M). Also, Hk := H2 k is a Hilbert space under the L2-inner product. F k contains only smooth functions. In general, a sequence in F k will not converge in the H k norm to a function in F k , so we need to complete the space to have anything useful. An alternate approach would have been to start with functions in Lp rather than completing the space of smooth functions ...

The Hodge theorem for Riemannian manifolds

Higman ’ s Embedding Theorem . An Elementary Proof

In 1961 G. Higman proved a remarkable theorem establishing a deep connection between the logical notion of recursiveness and questions about finitely presented groups. The basic aim of the present paper is to provide the reader with a rigorous and detailed proof of Higman’s Theorem. All the necessary preliminary material, including elements of group theory and recursive functions theory, is sys...

Gunther’s Proof of Nash’s Isometric Embedding Theorem

Around 1987 a German mathematician named Matthias Gunther found a new way of obtaining the existence of isometric embeddings of a Riemannian manifold. His proof appeared in [1, 2]. His approach avoids the so-called Nash-Moser iteration scheme and, therefore, the need to prove smooth tame or Moser-type estimates for the inverse of the linearized operator. This simplifies the proof of Nash’s isom...

A Proof of Perelman’s Collapsing Theorem for 3-manifolds

We will simplify the earlier proofs of Perelman’s collapsing theorem of 3-manifolds given by Shioya-Yamaguchi [SY00]-[SY05] and Morgan-Tian [MT08]. A version of Perelman’s collapsing theorem states that: “Let {M3 i } be a sequence of compact Riemannian 3-manifolds with curvature bounded from below by (−1) and diam(M3 i ) ≥ c0 > 0. Suppose that all unit metric balls in M3 i have very small volum...