A blowup proof of Aubin’s theorem in the Yamabe problem

There is a classic blowup analysis proof for Aubin’s theorem, due to Uhlenbeck’s renormalization method described below to give another proof that the Yamabe problem. See the chapter5 at Schoen & Yau              ‘ Lectures on differential geometry’.

Yamabe’s approach was to consider first the perturbed functional

\displaystyle  Q_s(u)\doteqdot\frac{\int_M\Big(|\nabla u|^2+\frac{n-2}{4(n-1)}R_gu^2\Big)d\mu_g}{\big(\int_M|u|^sd\mu_g\big)^{2/s}}

where s\in(0,\frac{2n}{n-2}] and u\in H^1(M)\setminus\{0\}. Set

\displaystyle  \lambda_s\doteqdot\inf\{Q_s(u):u\in H^{1}(M)\setminus\{0\}\}\quad\text{and}\; Y(M)=\lambda_{2^*}

By using a direct minimizing procedure, it can be shown that for 2 < s < 2^*, there exists a smooth positive function us such that its L^s-norm is equal to one, Q_s(u_s) = \lambda_s and us satisfies the equation

\displaystyle  \Delta_gu_s-\frac{n-2}{4(n-1)}R_gu_s+\lambda_su^{s-1}_s=0,\quad \text{in}\;M

The direct method does not work when s=2^* because the Sobolev embedding H^1(M)\to L^{2^*}(M)  is continuous but not compact.

However, if one can show that u_s is uniformly bounded, i.e.there exists a positive constant c such that u_s \le c in M for 2 < s < 2^*, then there exists a sequence \{s_i\} \in 2 < s < 2^* such that and u_{s_i} converges to a smooth positive function u which satisfies the Yamabe equation .

We discuss a blow-up argument. Suppose that no such upper bound c exists. It follows that there exist sequences \{s_k\} \subset (2, 2^*) and \{y_k\} \subset M such that

\displaystyle  s_k\to 2^*\quad\text{and}\quad m_k\doteqdot u_{s_k}(y_k)=\max u_{s_k}\to\infty,\quad as\;k\to\infty

As M is compact, we may assume that y_k \to y_0 as k \to\infty. For a normal coordinate system centered at y_0 and with radius \rho, let the coordinates of y_k be x_k, k = 1, 2, .....

In the local coordinates,

\displaystyle  g_{ij}(x)=\delta_{ij}+O(\rho^2),\quad \det g=1+O(\rho^2)

u_{k}=u_{s_k} satisfies the equation

\displaystyle  \frac1{\sqrt{\det g}}\partial_j\Big(\sqrt{\det g}g^{ij}\partial_iu_{k}\Big)-\frac{n-2}{4(n-1)}R_gu_{k}+\lambda{k}u^{{s_k}-1}_{k}=0,\quad \text{in}\;B_0(\rho)

The idea here is to consider the normalized function

\displaystyle  v_k\doteqdot\frac{u(\delta_kx+x_k)}{m_k}

where \delta_k=m_k^{(2-s_k)/2}. We have x_k \to 0 and \delta_k \to 0 as k \to\infty. Here v_k is defined on a ball in \mathbb{R}^n of radius \rho_k = (\rho-|x_k|)/\delta_k and \rho_k\to\infty as k\to\infty.

By the argument of diagonal subsequence and the property of normal coordinates , one observes that a subsequence of \{v_k\} converges to a smooth positive function v which is a nonnegative solution of the equation

\displaystyle  \Delta_0 v+\lambda v^{\frac{n+2}{n-2}}=0,\quad\text{in}\;\mathbb{R}^n\hfill (1)

where \lambda=\lim\limits_{k\to\infty}\lambda_k,and \Delta_0 is the standard Laplacian on \mathbb{R}^n.

By the strong maximum principle, v>0. It is known that \lambda<\lambda(M) if \lambda(M) < 0; and \lambda=\lambda(M) if \lambda(M) \ge 0 . Let d be the diameter of (M, g). By a change of variables we have

\displaystyle  \int_{|x|<\frac d2\delta_k^{-1}}v_k^{s_k}\sqrt{\det g}dx=\delta_k^{\frac{2s_k}{s_k-2}-n}\int_{B_{x_k}(\frac d2)}u_k^{s_k}d\mu_g\le\delta_k^{\frac{2s_k}{s_k-2}-n}\hfill (2)

where B_{x_k}(d/2) denotes the open ball in (M, g) with center at x_k and radius equal to d/2. we note that

 \displaystyle  \frac{2s_k}{s_k-2}-n>0\quad \text{and}\;\to0\quad\text{as}\;k\to\infty.

From (2) the Fatou lemma and \lim\limits_{k\to\infty}\delta_k\to0 , we obtain

\displaystyle  \int_{\mathbb{R}^n}v^{\frac{2n}{n-2}}dx\le1\hfill(3)

A similar argument implies

\displaystyle  \int_{\mathbb{R}^n}|\nabla v|^2dx<\infty.

Let \eta\in C^{\infty}_0(\mathbb{R}^n) be a cutoff function satisfies \eta =1 in B_0(d) and  \eta =0 in \mathbb{R}^n\setminus B_0(2d)

Defined v_R(x)=\eta{\frac xR}v(x), then

\displaystyle  \int_{\mathbb{R}^n}(|\nabla(v-v_R)|^2+|v-v_R|^{2^*})dx\to0,\quad \text{as}\;R\to\infty.\hfill (4)

Multiplies (1) by v_R and integration by parts, we obtain

\displaystyle  \int_{\mathbb{R}^n}\nabla v_R\nabla vdx=\lambda\int_{\mathbb{R}^n}v^{2^*-1}v_Rdx

Taking R\to\infty in above equation and thanks to (4) we get

\displaystyle  \int_{\mathbb{R}^n}|\nabla v|^2dx=\lambda\int_{\mathbb{R}^n}v^{2^*}dx.\hfill(5)

  •  If \lambda\le0, then v=\text{constant}, and (2)  implies v\equiv0, which is a contradiction with v>0.
  •  If \lambda>0, \lambda=\lambda(M). (2) (5) and the best Sobolev imbedding implies

 \displaystyle \Lambda\Big(\int_{\mathbb{R}^n}v^{2^*}dx\Big)^{2/2^*}\le\int_{\mathbb{R}^n}|\nabla v|^2dx=\lambda(M)\int_{\mathbb{R}^n}v^{2^*}dx.

 Thus

\displaystyle  \Lambda\le\lambda(M)\Big(\int_{\mathbb{R}^n}v^{2^*}dx\Big)^{n/2}\le\lambda(M).

  We are led to the contradiction with \lambda(M)<\lambda(\mathbb{S}^n)=\Lambda.
 Therefore, u_s is uniformly bounded.

发表在 Geometry, Mathematics | 标签为 , | 2条评论

A simple proof of Aubin’s theorem in Yamabe problem

Theorem(Aubin) Let (M, g) be a compact Riemannian manifold with Y(M, g) < Y (S_n), where Y(M, g) is called the Yamabe invariant and defined by

\displaystyle  Y(M,g)=\inf_{u\in C^{\infty}(M)}Q_g(u)=\inf_{ u\in C^{\infty}(M)}\frac{\int_M(|\nabla u|^2+\frac{n-2}{4(n-1)}R_gu^2)d\mu_g}{\|u\|^2_{L^{2^*}}}.

Then the infimum of the functional Q_g(u) is attained. Namely, the Yamabe problem can be solved.

The original proof of Theorem used the subcritical equations. There is another simple proof by Druet and Hebey using the  Brezis and Lieb’s lemma.

Proof: After passing to a subsequence, we may also assume that there exists u\in H^1(M) such that u_i\rightharpoonup u weakly in H^1(M), applying the imbedding theorem, we obtain u_i\to u strongly in L^2(M), furthermore u_i\to u almost everywhere as i\to\infty. In particular, u is nonnegative. It easy form the weakly convergence that

\displaystyle  \|\nabla u_i\|^2_{L^2}=\|\nabla(u_i-u)\|_{L^2}^2+\|\nabla u\|^2_{L^2}+o(1)

for all i, where o(1)\to0 as i\to\infty. We also have that(p=2^*=\frac{2n}{n-2}) by  Brezis and Lieb’s lemma

\displaystyle  \|u_i\|^p_{L^p}=\|u_i-u\|_{L^p}^p+\|u\|_{L^p}^p+o(1).

Thanks to the sharp Sobolev inequality of Hebey and Vaugon, there exists B>0 such that for any i,

\displaystyle  \|u_i-u\|^2_{L^p}\le K_n^2\|\nabla(u_i-u)\|^2_{L^2}+B\|u_i-u\|^2_{L^2}.

Since u_i\in \mathcal{H}=\{u\in H^1(M):\int_M|u|^pd\mu_g=1\}, it follows that

\displaystyle  (1-\|u\|^p_{L^p})^{\frac 2p}\le K_n^2\left(\|\nabla u_i\|^2_{L^2}-\|\nabla u\|^2_{L^2}\right)+o(1).

Since Q_g(u_i)\to Y_g(M), and since u_i\to u in L^2(M), we also have that

\displaystyle\begin{gathered}  K_n^2\left(\|\nabla u_i\|^2_{L^2}-\|\nabla u\|^2_{L^2}\right)\hfill\\  \qquad=K_n^2Y(M)-K_n^2\left(\int_M|\nabla u|^2d\mu+\int_MaRu^2d\mu\right)+o(1)\hfill\\  \qquad\le K_n^2Y(M)-K_n^2Y(M)\|u\|^2_{L^p}+o(1)\hfill\\  \end{gathered}

Hence,

\displaystyle  (1-\|u\|^p_{L^p})^{\frac 2p}\le K_n^2Y_g(M)(1-\|u\|^2_{L^p})+o(1).

We assume that Y_g(M)<Y_g(S^n)=1/K_n^2 and note that

\displaystyle  1-\|u\|^2_{L^p}\le (1-\|u\|^p_{L^p})^{\frac 2p}

this implies that

\displaystyle  \|u\|^2_{L^p}=1.

Then,

\displaystyle  \|\nabla u_i\|_{L^2}\to \|\nabla u\|_{L^2}

as i\to\infty. and since

\displaystyle  \|\nabla u_i\|^2_{L^2}=\|\nabla(u_i-u)\|_{L^2}^2+\|\nabla u\|^2_{L^2}+o(1)

We obtain that u_i\to u strongly in H^1 as i\to\infty. In particular, u is a minimizer for Y_g(M) and u is a weak nonnegative solution of the Yamabe equation

\displaystyle  \Delta_gu-\frac{n-2}{4(n-1)}R_gu=Y_g(M)u^{2^*-1}.

Regularity argument and the maximum principle then give that u is smooth and positive. This prove the Theorem.

发表在 Geometry, Mathematics | 标签为 , , | 2条评论

Ricci flow–a nonlinear PDE perspective

The important thing in life is to have a great aim, and dermination to attain it. ——Johan wolfgang von Goethe.

There is an elegant and profound paper by Terence Tao. In that paper the author discusses some of the key ideas of Perelman’s proof of Poincare’s conjecture via the Hamilton program of using the Ricci flow, from the perspective of the modern theory of nonlinear partial differential equations.

As the author of the article suggests a large part of Perelman’s work is actually conducted in the arena of nonlinear PDE arguments, would already be the most technically impressive and significant result in the field of nonlinear PDE in recent years; the fact that this PDE result also gives the Poincar´e conjecture and the more general geometrization conjecture makes it the best piece of mathematics we have seen in the last ten years. It is truly a landmark achievement for the entire discipline.

There are three excellently detailed expositions of Perelman’s work Kleiner-Lott , Morgan-Tian and Cao-Zhu , in addition to Perelman’s original papers 1 2 3. The author suggests that reading all these papers in parallel (in particular, switching from one paper to another whenever we were stuck on a particular point) we were able to obtain a far richer view of the argument than we could have obtained just from reading one of them. At last, the author’s lecture notes were very heuristic and valuable.

发表在 Probability | 标签为 , , | 留下评论