Mathematics

Featured

Posted on by
Reply

This therefore, is mathematics: she reminds you of the invisible forms of the soul; she gives life to her own discoveries; she awakens the mind and purifies the intellect; she brings to light our intrinsic ideas; she abolishes oblivion and ignorance which are ours by birth…

Steve Jobs(1955-2011)-Stay Hungry. Stay Foolish.

Posted on by
Reply

 

Trust-Don’t settle-follow your heart: Three stories from Jobs’s life

The first story is about connecting the dots.

Again, you can’t connect the dots looking forward; you can only connect them looking backwards. So you have to trust that the dots will somehow connect in your future. You have to trust in something – your gut, destiny, life, karma, whatever, because believing that the dots that will connect down the road will give you the confidence to follow your heart, even when it leads you off the well-worn path, and that will make all the difference.

My second story is about love and loss.

Sometimes life hits you in the head with a brick. Don’t lose faith. I’m convinced that the only thing that kept me going was that I loved what I did. You’ve got to find what you love. And that is as true for your work as it is for your lovers. Your work is going to fill a large part of your life, and the only way to be truly satisfied is to do what you believe is great work. And the only way to do great work is to love what you do. If you haven’t found it yet, keep looking. Don’t settle. As with all matters of the heart, you’ll know when you find it. And, like any great relationship, it just gets better and better as the years roll on. So keep looking until you find it. Don’t settle.

 

My third story is about death.

If you live each day as if it was your last, someday you’ll most certainly be right. If today were the last day of my life, would I want to do what I am about to do today? And whenever the answer has been “No” for too many days in a row, I know I need to change something.

Remembering that I’ll be dead soon is the most important tool I’ve ever encountered to help me make the big choices in life. Because almost everything – all external expectations, all pride, all fear of embarrassment or failure – these things just fall away in the face of death, leaving only what is truly important. Remembering that you are going to die is the best way I know to avoid the trap of thinking you have something to lose. You are already naked. There is no reason not to follow your heart.

Your time is limited, so don’t waste it living someone else’s life. Don’t be trapped by dogma – which is living with the results of other people’s thinking. Don’t let the noise of other’s opinions drown out your own inner voice. And most important, have the courage to follow your heart and intuition. They somehow already know what you truly want to become. Everything else is secondary

Stay Hungry. Stay Foolish. And I have always wished that for myself.

Compact gradient Yamabe solitons

Posted on by
Reply

Self-similar solutions and translating solutions, often called soliton solutions, have emerged in recent years as important objects in geometric flow since they appear possible singularity models. we are interested in geometric structure of Yamabe flow.

A complete Riemannian metric g on a smooth manifold M^n is called a gradient Yamabe soliton if there exists a smooth function f such that its Hessian satisfies the equation

\displaystyle \hfill\nabla_i\nabla_jf=(R-\rho)g_{ij},\hfill(1)

where R is the scalar curvature of g and \rho is a constant. For \rho=0,>0 and <0 the Yamabe soliton is steady, shrinking and expanding.

Theorem: If (M^n,g,f) is a compact Yamabe soliton, not necessarily locally conformally flat, then g is the metric of constant scalar curvature.

proof: Tracing the soliton equation yields

\displaystyle \Delta f=n(R-\rho).\hfill(2)

Applying \nabla_k to soliton equation, we obtain

\displaystyle \nabla_k\nabla_i\nabla_jf=\nabla_kRg_{ij}

implying that

\displaystyle \nabla_i\nabla_k\nabla_jf+R^j_{ki\ell}\nabla^{\ell}f=\nabla_kRg_{ij}

and tracing the previous equation in k and j

\displaystyle \nabla_i\Delta f+R_{i\ell}\nabla^{\ell}f=\nabla_iR.

By trace soliton equation, we get

\displaystyle n\nabla_iR+R_{i\ell}\nabla^{\ell}f=\nabla_iR.

Thus

(n-1)\nabla_iR+R_{i\ell}\nabla^{\ell}f=0.\hfill(3)

And taking divergence,

\displaystyle (n-1)\Delta R+\nabla^iR_{ij}\nabla^jf+\nabla^i\nabla^jfR_{ij}=0.

The contracted Bianchi identity and soliton equation imply

\displaystyle (n-1)\Delta R+\frac12\nabla_jR\nabla^jf+(R-\rho)g^{ij}R_{ij}=0

namely

\displaystyle (n-1)\Delta R+\frac12\langle\nabla R,\nabla f\rangle+(R-\rho)R=0.\hfill(4)

Taking Integral over trace soliton equation

\displaystyle \int_M(R-\rho)d\mu_g=\frac1n\int_M\Delta fd\mu_g=0.

By identity (4)

\displaystyle\begin{gathered} \int_MR(R-\rho)d\mu_g=\frac12\int_M\langle\nabla R,\nabla f\rangle d\mu_g=\frac12\int_MR\Delta fd\mu_g\hfill\\ \quad=\frac n2\int_MR(R-\rho)d\mu_g \end{gathered}

Since n\ge3, we have

\displaystyle \int_MR(R-\rho)d\mu_g=0,

and

\displaystyle \int_M(R-\rho)^2d\mu_g=0.

Therefore,

\displaystyle R=\rho=constant.

Scalar curvature equation on warped products manifolds

Posted on by
Reply

In this aritcle by Dobarro and Dozo, the authors give another method calculating the scalar curvatrue on warped products manifolds by making using of conformal change.

Let M = (M^m,g) and N = (N^n,h) be two Riemann manifolds. For \phi\in C^{\infty}(M), \phi> 0 on M, we consider the warped product

\displaystyle M\times_{\phi} N = ((M\times N)^{m+n}, g + \phi^2 h)

and show the relationship between the scalar curvatures R_g on M, R_h on N and R_{\phi} on M \times_{\phi} N. This relationship is a nonlinear partial differential equation satisfied by a power of the weight \phi.

Theorem:  Let R_g, R_h and R_{\phi} denote the scalar curvature on M, N and M \times_{\phi} N respectively. Then the following equality holds:

\displaystyle -\frac{4n}{n+1}\Delta_gu+R_gu+R_hu^{\frac{n-3}{n+1}}=R_{\phi}u\hfill (1)

where u=\phi^{\frac{n+1}2}. Namely,

\displaystyle R_{\phi}=R_g+\frac{R_h}{\phi^2}-2n\frac{\Delta\phi}{\phi}-n(n-1)\frac{|\nabla_g\phi|^2}{\phi^2}.

Proof: Write g_{\phi}\doteqdot g+\phi^2h=\phi^2(\phi^{-2}g+h), so g_{\phi} is conformal to \tilde{g}_{\phi}\doteqdot\phi^{-2}g+h on M\times N and \phi^{-2}g is conformal to g on M.

Suppose m \ge 3, we apply Yamabe equation in M to obtain that \phi satisfies

\displaystyle -\frac{4(m-1)}{m-2}\Delta_gv+R_gv=v^{\frac{n+2}{n-2}}\widetilde{R}_g\hfill (2)

with \phi^2=v^{\frac4{m-2}}, where \widetilde{R}_g denotes the scalar curvature on (M^m,\phi^{-2}g).
As m+n\ge3, we use Yamabe equation in M\times N. Hence \phi also satisfies

\displaystyle -\frac{4(m+n-1)}{m+n-2}\Delta_{\tilde{g}_{\phi}}w+\widetilde{R}_{g_{\phi}}w={R}_{g_{\phi}}w^{\frac{m+n+2}{m+n-2}}\hfill (3)

with w^{\frac4{m+n-2}}=\phi^2 where \widetilde{R}_{g_{\phi}} denotes the scalar curvature on (M^m+N^n,\tilde{g}_{\phi}) and \Delta_{\tilde{g}_{\phi}} the corresponding laplacian.

From w\in C^{\infty}(M), we deduce that

\displaystyle \Delta_{\tilde{g}_{\phi}}w=\Delta_{\phi^{-2}g+h}w=\Delta_{\phi^{-2}g}w.

Working in local coordinates

\displaystyle \Delta_{\phi^{-2}g}w=\frac1{\sqrt{|\phi^{-2}g|}}\partial_i\left((\phi^{-2}g)^{ij}\sqrt{|\phi^{-2}g|}\partial_jw\right)

where |\phi^{-2}g|=\det(\phi^{-2}g_{ij})=\phi^{-2m}|g| and (\phi^{-2}g)^{ij}=\phi^2g^{ij}, hence

\displaystyle\begin{gathered} \Delta_{\phi^{-2}g}w=\phi^m\frac1{\sqrt{|g|}}\partial_i\left(\phi^{2-m}g^{ij}\sqrt{|g|}\partial_jw\right)\hfill\\ \qquad=v^{\frac{-2m}{m-2}}\frac1{\sqrt{|g|}}\partial_i\left(v^2g^{ij}\sqrt{|g|}\partial_jw\right)\hfill\\ \qquad=\left(v\Delta_gw+2g^{ij}\partial_iv\partial_jw\right)v^{-\frac{n+2}{n-2}}. \end{gathered}

On the other hand,

\displaystyle \Delta_g(vw)=v\Delta_gw+w\Delta_gv+2g^{ij}\partial_iv\partial_jw.

Hence

\displaystyle v^{\frac{n+2}{n-2}}\Delta_{\phi^{-2}g}w=\Delta_g(vw)-w\Delta_gv.\hfill(4)

We have

\displaystyle \widetilde{R}_{g_{\phi}}=\widetilde{R}_g+R_h\hfill(5)

By using of this in (3)  and multiplying by v^{\frac{n+2}{n-2}}, we obtain from (4)

\displaystyle -\frac{4(m+n-1)}{m+n-2}\big(\Delta_g(vw)-w\Delta_gv\big)+v^{\frac{n+2}{n-2}}(\widetilde{R}_g+R_h)={R}_{g_{\phi}}w^{\frac{m+n+2}{m+n-2}}v^{\frac{n+2}{n-2}}\hfill (6)

From (2) we arrive at

\displaystyle\begin{gathered} -\frac{4(m+n-1)}{m+n-2}\Delta_g(vw)-\frac{4n}{(m+n-2)(m-2)}w\Delta_gv\hfill\\ \qquad+R_g(vw)+R_hwv^{\frac{n+2}{n-2}}=R_{\phi}(vw)w^{\frac4{m+n-2}}v^{\frac4{m-4}}=R_{\phi}(vw). \end{gathered}

Recalling that w^{\frac4{m+n-2}}v^{\frac4{m-4}}=\phi^2\phi^{-2}=1 and denoting u =\phi^{\frac{n+1}2}, replacing in terms of u in this last equality, then multiplying by u^{1/n+1}, we obtain

\displaystyle\begin{gathered} -\frac{4(m+n-1)}{m+n-2}\Delta_g(u^{\frac n{n+1}})u^{\frac1{n+1}}-\frac{4n}{(m+n-2)(m-2)}u^{\frac{m+n-1}{n+1}}\Delta_gu^{\frac{2-m}{n+1}}\hfill(7)\\ \quad+R_gu+R_hu^{\frac{n-3}{n+1}}=R_{\phi}u. \end{gathered}

For any \alpha\neq0, u\in C^{\infty}(M), u>0 such that

\displaystyle \Delta_gu^{\alpha}=\alpha(\alpha-1)u^{\alpha-2}|\nabla_gu|^2+\alpha u^{\alpha-1}\Delta_gu.

Choosing \alpha=\frac n{n+1} and \frac{2-m}{n+1} in (7), we obtain

\displaystyle -\frac{4n}{n+1}\Delta_gu+R_gu+R_hu^{\frac{n-3}{n+1}}=R_{\phi}u.

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

Posted on by
3

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.

A simple proof of Aubin’s theorem in Yamabe problem

Posted on by
2

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.

Yamabe Flow

Posted on by
Reply

The Yamabe conjecture states that given a compact Riemannian manifold (M, g), there exists a metric g pointwise conformal to g with constant scalar curvture R. In 1984, Schoen obtained a complete solution to the Yamabe conjecture.

Recently, the flow techniques display enormous power in soloving many great problems, especially for Poincare conjecture and sphere surface theorem. Thus, there is every reason to believe that we can apply these remarkable techniques to solve other problems or give alternate proof for interesting questions.

A nutrual idea is to find a evolution equation which deforms any Riemannian metric conformally to a constant scalar curvatue metric. That is Yamabe flow, it was introduced by Hamilton in 1988 after Ricci flow, as an approach to solve the Yamabe problem on manifolds of positive conformal Yamabe invariant. Yamabe flow is given by the following parabolic PDE

\displaystyle \partial_t g(x, t)=(r(t)-R(x,t))g(x, t), \ \ \ g(0, x)=g_0(x),

for x\in M and t\ge 0 , where r(t)=\frac{\int_M RdV}{\int_MdV} is the average scalar curture of metric g(t).

Yamabe flow is the negative L^2 gardient flow of the (normalized) total scalar curvture restricted to a given conformal class, it can be interpreted as deforming a Riemannian metric to a conformal metric of constant scalar curvature, when this flow converges.

The details see as the papers of Hamilton, Chow, Ye, Struwe and Brendle…

Singular Yamabe problems

Posted on by
Reply

One of my research theme of this term was about the asymptotic behaviour of solution to some semilinear PDE with singualrities , the first paper I read was by Richard Schoen at Invent. in 1999.

They consider the asymptotic behaviour of positive solutions u of the conformal scalar curvature equation,

\displaystyle \Delta u+\frac{n(n-2)}4 u^{\frac{n+2}{n-2}}=0,

in the neighbourhood of isolated singularities in the standard Euclidean ball. They present a much simpler and more geometric derivation of the asymptotic radial symmetry for such solutions.

They also discuss a refinement, showing that any such solution is asymptotic to one of the deformed radial singular solutions. Finally they give some applications of these refined asymptotics, first to computing the global Pohozaev invariants of solutions on the sphere with isolated singularities, and then to the regularity of the moduli space of all such solutions.