Ngô Quốc Anh

December 15, 2010

The Cheeger isoperimetric constant

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 19:30

In Riemannian geometry, the Cheeger isoperimetric constant of a compact Riemannian manifold M is a positive real number h(M) defined in terms of the minimal area of a hypersurface that divides M into two disjoint pieces of equal volume. In 1970, Jeff Cheeger proved an inequality that related the first nontrivial eigenvalue of the Laplace-Beltrami operator on M to h(M). This proved to be a very influential idea in Riemannian geometry and global analysis and inspired an analogous theory for graphs.

The Cheeger constant. Let M be an n-dimensional closed Riemannian manifold. Let V(A) denote the volume of an n-dimensional submanifold A and S(E) denotes the n-1-dimensional volume of an submanifold E (commonly called “area” in this context).

Definition. The Cheeger isoperimetric constant of M is defined as

\displaystyle h(M)=\inf_E \frac{S(E)}{\min(V(A), V(B))},

where the infimum is taken over all smooth n-1-dimensional submanifolds E of M which divide it into two disjoint submanifolds with boundary A and B. Isoperimetric constant may be defined more generally for noncompact Riemannian manifolds of finite volume.

It is well-known that the Cheeger constant h(M) can be characterized by the following

\displaystyle h(M): = \mathop {\inf }\limits_{f \in C_0^\infty  (M)\backslash \{ 0\} } \frac{{\int_M {\left| {\nabla f} \right|d\mu }  }}{{\int_M {\left| f \right|d\mu } }}.

The Cheeger inequality. The Cheeger constant h(M) and \lambda_1(M), the smallest positive eigenvalue of the Laplacian on M, are related by the following fundamental inequality proved by Jeff Cheeger

\displaystyle \lambda_1(M)\geq \frac{h^2(M)}{4}.

This inequality is optimal in the following sense: for any h>0, natural number k and \varepsilon>0, there exists a two-dimensional Riemannian manifold M with the isoperimetric constant h(M) = h and such that the kth eigenvalue of the Laplacian is within \varepsilon from the Cheeger bound (Buser, 1978).

Following is the proof of the Cheeger inequality. I found this proof from a book due to Alexander Grigoryan (problem 10.8).

Proof. Replacing f by f^2 we obtain

\displaystyle h(M) \leqslant \frac{{\int_M {\left| {\nabla ({f^2})} \right|d\mu } }}{{\int_M {{f^2}d\mu } }} = 2\frac{{\int_M {\left| f \right|\left| {\nabla f} \right|d\mu } }}{{\int_M {{f^2}d\mu } }} \leqslant 2\frac{{{{\left\| f \right\|}_{{L^2}}}{{\left\| {\nabla f} \right\|}_{{L^2}}}}}{{\left\| f \right\|_{{L^2}}^2}} = 2\frac{{{{\left\| {\nabla f} \right\|}_{{L^2}}}}}{{{{\left\| f \right\|}_{{L^2}}}}}.

Taking \inf in f, we obtain the desired inequality.

The Buser inequality. Peter Buser proved an upper bound for \lambda_1(M) in terms of the isoperimetric constant h(M). Let M be an n-dimensional closed Riemannian manifold whose Ricci curvature is bounded below by -(n-1)a^2, where a \geqslant 0. Then

\displaystyle\lambda_1(M)\leq 2a(n-1)h(M) + 10h^2(M).

For a proof of the Buser inequality, we refer the reader to a paper due to Ledoux published in Proc. Amer. Math. Soc. [here].

Source: WikiPedia

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December 11, 2010

The Positive Energy Theorem

Filed under: Riemannian geometry — Ngô Quốc Anh @ 17:01

In general relativity, the positive energy theorem (more commonly known as the positive mass theorem in differential geometry) states that, assuming the dominant energy condition, the mass of an asymptotically flat spacetime is non-negative; furthermore, the mass is zero only for Minkowski spacetime. The theorem is a scalar curvature comparison theorem, with asymptotic boundary conditions, and a corresponding statement of geometric rigidity.

The proof. The original proof of the theorem for ADM mass was provided by Richard Schoen and Shing-Tung Yau in 1979 using variational methods. Edward Witten gave a simpler proof in 1981 based on the use of spinors, inspired by positive energy theorems in the context of supergravity. An extension of the theorem for the Bondi mass was given by Ludvigsen and James Vickers, Gary Horowitz and Malcolm Perry, and Schoen and Yau.

Gary Gibbons, Stephen Hawking, Horowitz and Perry proved extensions of the theorem to asymptotically anti-de Sitter spacetimes and to Einstein–Maxwell theory. The mass of an asymptotically anti-de Sitter spacetime is non-negative and only equal to zero for anti-de Sitter spacetime. In Einstein–Maxwell theory, for a spacetime with electric charge Q and magnetic charge P, the mass of the spacetime satisfies

\displaystyle M \geq \sqrt{Q^2 + P^2},

with equality for the Majumdar–Papapetrou extremal black hole solutions.

(more…)

December 8, 2010

A geometric description of the neighbourhood of a critical point given by the mountain-pass theorem

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 12:35

Recently, I have read a paper by H. Hofer published in J. London Math. Soc. [here]. In that paper the author studies the behaviour of a functional in the neighborhood of its critical points given by the mountain-pass theorem.

Let us start with the following notations. Assume F is a real Banach space and that \Phi \in C^1(F, \mathbb R). For real numbers c and d, we first define the set of critical points with level set d

\displaystyle Cr(\Phi ,d) = \left\{ {u \in F:\Phi (u) = d,\Phi '(u) = 0} \right\}.

Then we define

\displaystyle Cr(\Phi ) = \bigcup\limits_{d \in \mathbb{R}} {Cr(\Phi ,d)} .

Next we define

\displaystyle\begin{gathered}  {\Phi ^d} = {\Phi ^{ - 1}}(( - \infty ,d]), \hfill \\ {{\dot \Phi }^d} =  {\Phi ^{ - 1}}(( - \infty ,d)), \hfill \\ {\Phi _c} = {\Phi ^{ -  1}}([c, + \infty )), \hfill \\ \Phi _c^d = {\Phi ^d} \cap {\Phi _c}.  \hfill \\ \end{gathered}

We say that \Phi satisfies the Palais-Smale condition if the following holds.

If for some sequence \{u_n\} \in F we have \Phi(u_n) \to d \in \mathbb R and \Phi'(u_n) \to 0 in F^\star then \{u_n\} is precompact.

Definition. Let \Phi \in C^1(F, \mathbb R) and assume that u_0 \in Cr(\Phi,d). Then

  1. u_0 is a local minimum if there exists an open neighbourhood V of u_0 such that \Phi(u) \geqslant \Phi(u_0) for all u \in V;
  2. u_0 is of mountain-pass type if for all open neighbourhoods U of u_0, {\dot \Phi }^d \cap U\ne \emptyset and {\dot \Phi }^d \cap U is not path connected.

Following is the main result of the Hofer paper.

Theorem. Let \Phi \in C^1(F, \mathbb R) satisfy Palais-Smale condition and assume that e_0, e_1 are distinct points in F. Define

\displaystyle A = \left\{ {a \in C([0,1],F):a(i) = {e_i},i = 0,1} \right\}

and

\displaystyle d = \mathop {\inf }\limits_{a \in A} \sup \Phi (a([0,1])), \quad c = \max \left\{ {\Phi ({e_0}),\Phi ({e_1})} \right\}.

Then if d> c the set Cr(\Phi, d) is non-empty. Moreover there exists at least one critical point u_0 in Cr(\Phi, d) which is either a local minimum or of mountain-pass type. If all the critical points in Cr (\Phi, d) are isolated in F the set Cr (\Phi, d) contains a critical point of mountain-pass type.

The proof is based on a topological lemma and on the deformation lemma and we leave it for interested reader.

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