Posted by: matheuscmss | February 28, 2015

## First Bourbaki seminar of 2015 (II): Carron’s talk

For the second installment of this series of posts (which started here) on the first Bourbaki seminar of 2015, we will discuss Gilles Carron talk entitled “New utilisation of the maximum principles in Geometry (after B. Andrews, S. Brendle, J. Clutterbuck)”. Here, besides the original works of Andrews-Clutterbuck and Brendle (quoted below), the main references are the video of Carron’s talk and his lecture notes (both in French).

Disclaimer. All errors, mistakes or misattributions are my entire responsibility.

1. Introduction

Given a Riemannian ${n}$-dimensional manifold ${M}$, one can often study its Geometry by analyzing adequate smooth real functions ${f}$ on ${M}$ (such as scalar curvature). One of the techniques used to get some information about ${f}$ is the following observation (“baby maximum principle”): if ${f}$ has a local maximum at a point ${p}$, then we dispose of

• a first order information: the gradient of ${f}$ at ${p}$ vanishes; and
• a second order information: the Hessian of ${f}$ at ${p}$ has a sign (namely, it is negative definite).

In order to extract more information from this technique, one can appeal to the so-called doubling of variables method: instead of studying ${f}$, one investigates the local maxima of a “well-chosen” function ${g}$ on the double of variables (e.g., ${g:M\times M\rightarrow\mathbb{R}}$). In this way, we have new constraints because the gradient and Hessian of ${g}$ depend on more variables than those of ${f}$.

This idea of doubling the variables goes back to Kruzkov who used it to estimate the modulus of continuity of the derivative of solutions of a non-linear parabolic PDE (in one space dimension). In this post we shall see how this idea was ingeniously employed by Andrews and Clutterbuck (2011) and Brendle (2013) in two recent important works.

Theorem 1 (Andrews-Clutterbuck) Let ${\Omega\subset\mathbb{R}^n}$ be a convex domain of diameter ${D}$. Consider the Schrödinger operator ${-\Delta+V}$ where ${\Delta=\sum\limits_{i=1}^n\frac{\partial^2}{\partial^2 x_i}}$ is the Laplacian operator and ${V}$ is the operator induced by the multiplication by a convex function ${V:\Omega\rightarrow\mathbb{R}}$.Recall that the spectrum of ${-\Delta+V}$ with respect to Dirichlet condition on the boundary ${\partial\Omega}$ consists of a discrete set of eigenvalues of the form: ${\lambda_1<\lambda_2\leq \dots}$

In this setting, the fundamental gap ${\lambda_2-\lambda_1}$ of ${-\Delta+V}$ is bounded from below by

$\displaystyle \lambda_2 - \lambda_1 \geq 3\frac{\pi^2}{D^2}$

Remark 1 This theorem is sharp: ${\lambda_2-\lambda_1=3\frac{\pi^2}{D^2}}$ when ${\Omega=(-D/2, D/2)\subset\mathbb{R}}$ and ${V\equiv 0}$ (by Fourier analysis). In other terms, Andrews-Clutterbuck theorem is an optimal comparison theorem between the fundamental gap of general Schrödinger operators with the one-dimensional case.

Next, we state Brendle’s theorem:

Theorem 2 (Brendle) A minimal torus inside the round sphere ${S^3=\{(x_1,\dots, x_4\in\mathbb{R}^4: x_1^2+\dots+x_4^2=1\}}$ is isometric to Clifford torus ${\mathbb{T}=\{(x_1,\dots,x_4)\in\mathbb{R}^4: x_1^2+x_2^2 = x_3^2+x_4^2 = 1/2\}}$.

The sketches of proof of these results are presented in the next two Sections. For now, let us close this introductory section by explaining some of the motivations of these theorems.

1.1. The context of Andrews-Clutterbuck theorem

The interest of the fundamental gap ${\gamma=\lambda_2-\lambda_1}$ comes from the fact that it helps in the description of the long-term behavior of non-negative non-trivial solutions of the heat equation

$\displaystyle \frac{\partial}{\partial t} u(t,x) = \Delta u(t,x) - V(x)u(t,x), \quad (t,x)\in [0,\infty)\times \Omega$

with ${u\equiv 0}$ on ${\partial \Omega}$. More precisely, one has that

$\displaystyle u(t,x) = c \exp(\lambda_1t) f_1(x) (1+O(\exp(-\gamma t))$

where

• ${c}$ is an adequate constant,
• ${f_1}$ is the ground state of ${-\Delta+V}$, i.e., ${-\Delta f_1+V f_1=\lambda_1 f_1}$, ${f_1>0}$ on ${\textrm{int}(\Omega)}$, ${f_1=0}$ on ${\partial\Omega}$ and ${f_1}$ is normalized so that ${\int_{\Omega} f_1^2=1}$, and
• ${O(\exp(-\gamma t))}$ denotes (as usual) a quantity bounded from above by ${C\exp(-\gamma t)}$ for some constant ${C>0}$ and all ${t\geq 0}$.

The theorem of Andrews-Clutterbuck answers positively a conjecture of Yau and Ashbaugh-Benguria. This conjecture was based on a series of works in Mathematics and Physics: from the mathematical side, van den Berg observed during his study of the behavior of spectral functions in big convex domains (modeling Bose-Einstein condensation) that ${\lambda_2-\lambda_1\geq 3\pi^2/D^2}$ for the free Laplacian (${V\equiv 0}$) on several convex domains. After that, Singer-Wong-Yau-Yau proved that

$\displaystyle \lambda_2-\lambda_1\geq \frac{1}{4}\left(\frac{\pi^2}{D^2}\right)$

and Yu-Zhong improved this result by showing that

$\displaystyle \lambda_2-\lambda_1\geq \frac{\pi^2}{D^2}$

Furthermore, some particular cases of Andrews-Clutterbuck were previously known: for instance, Lavine proved the one-dimensional case ${\Omega\subset\mathbb{R}}$, and other authors studied the cases of convex domains with some (axial and/or rotational) symmetries in higher dimensions.

1.2. The context of Brendle theorem

The theorem of Brendle answers affirmatively a Lawson’s conjecture.

Lawson arrived at this conjecture after proving (in this paper here) that every compact oriented surface ${\Sigma}$ without boundary can be minimally embedded in ${S^3}$.

Remark 2 The analog of Lawson’s theorem is completely false in ${\mathbb{R}^3}$: using the maximum principle, one can show that there are no immersed compact minimal surfaces in ${\mathbb{R}^3}$.

Moreover, Lawson (in the same paper loc. cit.) showed that, if the genus of ${\Sigma}$ is not prime, then ${\Sigma}$ admits two non-isometric minimal embeddings in ${S^3}$.

On the other hand, Lawson’s construction in the case of genus ${1}$ produces only the Clifford torus (up to isometries). Nevertheless, Lawson proved (in this paper here) that if ${\Sigma\subset S^3}$ is a minimal torus, then there exists a diffeomorphism ${F:S^3\rightarrow S^3}$ taking ${\Sigma}$ to the Clifford torus ${\mathbb{T}}$: in other terms, there is no knotted minimal torus in ${S^3}$!

In this context, Lawson was led to conjecture that this diffeomorphism ${F:S^3\rightarrow S^3}$ could be taken to be an isometry, an assertion that was confirmed by Brendle.

2. Proof of Andrews-Clutterbuck theorem

One of the key points of Andrews-Clutterbuck argument is an improvement of a theorem of Brascamp-Lieb. More precisely, the Brascamp-Lieb theorem ensures, in the context of Theorem 1, the log-concavity of the the ground state ${f_1}$ of ${-\Delta+V}$ (i.e., the logarithm ${u=\log f_1}$ is a concave function). In this setting, a fundamental ingredient in Andrews-Clutterbuck proof of Theorem 1 is a quantitative statement about the log-concavity of ${f_1}$.

Before discussing Andrews-Clutterbuck’s improvement of Brascamp-Lieb theorem, let us quickly review Korevaar’s proof of Brascamp-Lieb theorem as an excuse to introduce a first concrete instance of the doubling of variables method.

2.1. A sketch of Korevaar’s proof of Brascamp-Lieb theorem

We want to show that ${f_1}$ is log-concave. For this sake, we can assume that the domain ${\Omega}$ and the potential ${V}$ are strictly convex. Indeed, this is so because ${\Omega}$ and ${V}$ are convex, so that they can be approximated by strictly convex objects, and, furthermore, it can be shown that the ground state ${f_1}$ varies continuously under deformations of ${\Omega}$ and ${V}$.

By definition, ${u(x)=\log f_1(x)}$ is concave if and only if the function

$\displaystyle Z(x,y) = u(x) + u(y) - 2u((x+y)/2)$

on the double of variables ${(x,y)\in\Omega\times\Omega}$ is non-positive.

We divide the proof of the fact that ${Z(x,y)\leq 0}$ for all ${(x,y)\in \Omega\times\Omega}$ into two parts.

First, we claim that ${\limsup\limits_{(x,y)\rightarrow\partial(\Omega\times\Omega)} Z(x,y)=0}$. In fact:

• If ${(x,y)\rightarrow (x_0,y_0)\in\partial(\Omega\times\Omega)}$ with ${x_0\neq y_0}$, then ${Z(x,y)\rightarrow -\infty}$ because ${f_1\equiv 0}$ (i.e., ${u=-\infty}$) on ${\partial\Omega}$ and ${f_1>0}$ on ${\Omega}$. Here, we used that ${(x_0+y_0)/2\in\Omega}$.
• If ${(x,y)\rightarrow (x_0,x_0)}$ with ${x_0\in\partial\Omega}$, one exploits the strict convexity of ${\Omega}$ to say that, near ${\partial\Omega}$, the ground state ${f_1}$ “looks like” the distance to the boundary ${\partial\Omega}$, so that ${u=\log f_1}$ is a concave function near ${\partial\Omega}$.

Next, once we dispose of the fact that ${\limsup\limits_{(x,y)\rightarrow\partial(\Omega\times\Omega)} Z(x,y)=0}$, the proof of the log-concavity of ${f_1}$ will be complete if we show that ${Z(x,y)=0}$ at any local maximum ${(x,y)\in \Omega\times\Omega}$.

In this direction, we use the baby maximum principle. If ${(x,y)\in\Omega\times\Omega}$ is a local maximum of ${Z}$, then ${Z}$ vanishes to the first order at ${(x,y)}$, i.e., ${dZ(x,y)=0}$. Thus, if denoting by ${m:=(x+y)/2}$, we deduce from the definition of ${Z}$ and the equation ${\nabla Z(x,y)=0}$ that

$\displaystyle du(x) = du(y) = du(m) \ \ \ \ \ (1)$

Moreover, by varying ${Z}$ in the direction of a small vector ${v}$, we get a function

$\displaystyle v\mapsto u(x+v) + u(y+v) - 2u(m+v)$

possessing a local maximum at ${v=0}$. Therefore, the Laplacian of this function at ${v=0}$ is non-positive, i.e.,

$\displaystyle \Delta u(x)+\Delta u(y)-2\Delta u(m)\leq 0 \ \ \ \ \ (2)$

Now, a simple calculation reveals that the Laplacian of ${u=\log f_1}$ satisfies the equation

$\displaystyle \Delta u = -\lambda_1 + V - |du|^2$

because ${f_1}$ is the ground state of ${-\Delta+V}$ (i.e., ${-\Delta f_1 + V f_1 = \lambda_1 f_1}$). Combining this equation with (1) and (2), we conclude that

$\displaystyle V(x)+V(y)-2\,V\left(\frac{x+y}{2}\right)\leq 0$

Since ${V}$ is strictly convex, this inequality implies that ${x=y}$, and, a fortiori, ${Z(x,y)=0}$, as we wanted to prove. This completes the sketch of Korevaar’s proof of Brascamp-Lieb theorem.

2.2. An improvement of Brascamp-Lieb’s theorem

The improvement of Andrews-Clutterbuck of the Brascamp-Lieb theorem consists of the following estimate of the modulus of continuity of the derivative of ${u=\log f_1}$:

$\displaystyle \left\langle\nabla\log f_1(x) - \nabla\log f_1(y),\frac{x-y}{\|x-y\|}\right\rangle \leq -2\frac{\pi}{D} \tan\left(\frac{\pi}{2}\frac{|x-y|}{D}\right) \ \ \ \ \ (3)$

This estimate provides new important informations beyond the statement of Brascamp-Lieb theorem: for example, when ${|x-y|\rightarrow D=\textrm{diam}(\Omega)}$, the right-hand side of the inequality goes to ${-\infty}$ (which is much better than simply knowing that it is non-positive).

The proof of this estimate is somewhat complicated: it involves a combination of the doubling of variables method, a comparison argument with the one-dimensional case and the study of parabolic PDEs.

For this reason, by following Carron’s talk, we will skip the proof of this estimate, and we will now discuss how this estimate can be used to get lower bound on the fundamental gap in Theorem 1. In this direction, we will follow an approach proposed by Lei Ni (which is not exactly the original argument of Andrews-Clutterbuck).

2.3. End of the (sketch of) proof of Theorem 1

We consider the eigenfunction ${f_2\neq 0}$ with ${-\Delta f_2 + V f_2 = \lambda_2 f_2}$, ${\int f_2^2=1}$, ${\int f_2 f_1 = 0}$, and ${f_2=0}$ on ${\partial\Omega}$.

The quotient ${g=f_2/f_1}$ is closely related to the fundamental gap ${\lambda_2-\lambda_1}$: more precisely, the function ${g}$ verifies

$\displaystyle \Delta g + 2\langle\nabla \log f_1, \nabla g\rangle + (\lambda_2-\lambda_1)g=0 \ \ \ \ \ (4)$

and ${\partial g/\partial\nu = 0}$ on ${\partial\Omega}$ (where ${\nu}$ is the unit outward normal to ${\partial\Omega}$).

The previous method of Singer-Wong-Yau-Yau consisted of studying first two derivatives of the function

$\displaystyle \frac{|dg|^2}{\|g\|_{L^{\infty}}^2-g^2}$

at its local maximum points, extract an inequality, and obtain a (non-optimal) lower bound on ${\lambda_2-\lambda_1}$ by integration of this inequality (together with the fact that ${g}$ satisfies (4)).

The approach proposed by Andrews-Clutterbuck consists in studying the oscillations of ${g}$, i.e., we compare ${g(x)-g(y)}$ with the one-dimensional model (on an interval) by means of the function:

$\displaystyle \mathcal{C}(x,y) = \frac{g(x)-g(y)}{2\sin(\alpha|x-y|/2)}$

where ${\alpha<\pi/D}$ and ${D=\textrm{diam}(\Omega)}$. (This choice of ${\alpha}$ avoids “boundary effects”).

We want to use the doubling of variables method, i.e., the baby maximum principle applied to ${\mathcal{C}(x,y)}$. For this sake, we introduce the quantity:

$\displaystyle \mu=\sup\limits_{(x,y)\in\Omega\times\Omega} \mathcal{C}(x,y).$

One can show that the strict convexity of ${\partial\Omega}$ implies that ${\mu}$ is attained on ${\textrm{Diag}:=\{(x,x): x\in\Omega\}}$ or in the interior of ${\Omega\times\Omega-\textrm{Diag}}$.

Remark 3 We have not defined ${\mathcal{C}}$ on ${\textrm{Diag}}$, but a first order expansion says that it is natural to pose

$\displaystyle \mathcal{C}(x,x):=|dg|(x)/\alpha$

In both cases, the study of the first two derivatives of ${\mathcal{C}(x,y)}$ at a maximum point and the improvement (3) of Brascamp-Lieb theorem imply that

$\displaystyle \lambda_2-\lambda_1\geq 3\alpha^2. \ \ \ \ \ (5)$

Of course, since ${\alpha<\pi/D}$ is arbitrary, this proves that ${\lambda_2-\lambda_1\geq 3(\pi/D)^2}$. Hence, the proof of Theorem 1 is complete once we prove (5).

For the sake of exposition, we show the validity of (5) only in the case that ${\mu}$ is attained at ${\textrm{Diag}}$, i.e., ${|dg|(x)/\alpha = \mathcal{C}(x,x) = \mu}$ for some ${x\in\Omega}$: indeed, the other case is similar (in some sense) to this one.

If ${\mu}$ is attained at ${(x,x)\in\textrm{Diag}}$,, then ${x}$ is a maximum for ${|dg|^2}$, so that

$\displaystyle \Delta |dg|^2(x)\leq 0$

Of course, this inequality is a good starting point to study ${\lambda_2-\lambda_1}$ (via (4)), but it is only a partial information obtained by letting ${(x,x)}$ vary only along ${\textrm{Diag}}$!

If we vary ${(x,x)}$ along the transverse direction by considering ${(x+v, x-v)}$ where ${v}$ is a small vector, we obtain from ${\mathcal{C}(x,x)=\mu}$ (and the baby maximum principle) that

$\displaystyle \Delta |dg|^2(x)\leq -2\alpha^2|dg|^2(x) \ \ \ \ \ (6)$

which is certainly a better estimate than the previous one.

In other words, we got an extra (better) information on ${dg}$ thanks to the doubling of variables method applied to ${\mathcal{C}(x,y)}$!

By differentiating the equation (4), and then applying (6) to the resulting PDE, we deduce that

$\displaystyle -(\lambda_2-\lambda_1)|dg|^2(x)-2 \textrm{Hess}\log f_1(x)(\nabla g(x),\nabla g(x))\leq -\alpha^2 |dg|^2(x)$

Now, we notice that the Andrews-Clutterbuck improvement (3) of Brascamp-Lieb theorem says (among other things) that ${\textrm{Hess}\log f_1(x)(\nabla g(x),\nabla g(x))\leq -(\pi/D)^2|dg|^2(x)}$. By plugging this into the previous inequality, we conclude that

$\displaystyle -(\lambda_2-\lambda_1)|dg|^2(x) - 2(\pi/D)^2|dg|^2(x)\leq -\alpha^2|dg|^2(x).$

Since ${g=f_2/f_1}$ is a non-constant function (and ${\alpha<\pi/D}$), we have from this inequality that

$\displaystyle \lambda_2-\lambda_1\geq \alpha^2+2(\pi/D)^2\geq 3\alpha^2$

This proves (5) when ${\mu}$ is attained at ${\textrm{Diag}}$, as desired.

3. Proof of Brendle theorem

Let ${\Sigma}$ be a minimal torus inside the round ${3}$-sphere ${S^3}$. Denote by ${\vec{\nu}:\Sigma\rightarrow S^3}$ a choice of unit normal to ${\Sigma}$.

The second fundamental form ${II_x:T_x\Sigma\rightarrow\mathbb{R}}$ is the Hessian at ${x}$ of the function ${h}$ (from ${T_x\Sigma}$ to ${\mathbb{R}\vec{\nu}(x)}$) whose graph over ${T_x\Sigma}$ is (locally) equal to ${\Sigma}$. In particular, ${II_x}$ is a symmetric quadratic form, and, hence, ${II_x}$ can be diagonalized. The (real) eigenvalues of ${II_x}$ are called principal curvatures of ${\Sigma}$ at ${x}$.

By definition, ${\Sigma}$ is minimal if and only if the trace of ${II_x}$ vanishes (for all ${x\in\Sigma}$). In other words, the eigenvalues of ${II_x}$ are ${\psi(x)\geq -\psi(x)}$ when ${\Sigma}$ is minimal.

For later use, we recall the following three facts:

• Lawson proved that ${\psi>0}$ when ${\Sigma}$ is a minimal torus in ${S^3}$. (Of course, this result strongly uses that ${\Sigma}$ has genus ${1}$, and, indeed, it is completely false for other genera)
• The minimality of ${\Sigma}$ imposes a constraint on ${II_x}$ known as Simon’s formula. In our setting, this means that the principal curvature ${\psi(x)}$ verifies the following PDE:

$\displaystyle \Delta\psi -\frac{|d\psi|^2}{\psi}+2(\psi^2-1)\psi = 0 \ \ \ \ \ (7)$

• Lawson also showed that if ${\psi}$ is constant, then the minimal torus ${\Sigma}$ is isometric to Clifford’s torus ${\mathbb{T}}$.

The last item above says that our objective is very clear: in order to prove Brendle’s theorem 2, we have to show that ${\psi}$ is constant.

By Gauss-Bonnet theorem, we have that ${\int_{\Sigma}\psi^2 = \textrm{area}(\Sigma)}$, i.e., ${\psi}$ equals to ${1}$ in average. From this point, a natural strategy would be to combine this information with Simons’ equation (and some maximum principles) to show that ${\max\psi\leq 1}$. Unfortunately, this idea does not work mainly because of the (negative) sign of the term ${|d\psi|^2/\psi}$ in Simons formula.

At this point, Brendle introduces the function

$\displaystyle \phi(x) = \sup\limits_{y\neq x} \frac{|\langle\vec{\nu}(x),y\rangle|}{1-\langle x,y\rangle}.$

(Note that, since ${x,y\in S^3}$, ${\|x\|=\|y\|=1}$, and, thus, ${\langle x,y\rangle = \textrm{dist}(x,y)^2}$.)

The geometrical meaning of ${\phi}$ is the following. The quantity ${1/\phi(x)}$ is the biggest radius ${R}$ such that ${\Sigma}$ stays outside a ball of radius ${R}$ tangent to ${\Sigma}$ at ${x}$.

In other terms, ${\Sigma}$ stays outside of the oscullating balls ${B^{\pm}_{x,R} = B(x\pm R\vec{\nu}(x), R)}$ of radius ${R}$, and ${\Sigma}$, ${B^+_{x,R}}$ and ${B^-_{x,R}}$ are mutually tangent at ${x}$. From this fact, it is possible to check that

$\displaystyle \phi(x)\geq\psi(x)$

This means that the global information ${\phi}$ (curvature of oscullating balls) controls the local information ${\psi}$ (principal curvature).

We affirm that the inequality ${\phi\geq\psi}$ implies that ${\phi}$ satisfies the following version of Simons formula

$\displaystyle \Delta\phi -\frac{|d\phi|^2}{\phi} + 2(\psi^2-1)\phi\geq 0 \ \ \ \ \ (8)$

in the sense of viscosity. For the sake of exposition, let us prove that ${\phi}$ satisfies this inequality when ${\phi\in C^{\infty}}$: the general case (${\phi}$ is a viscosity solution when ${\phi}$ is not smooth) follows by a simple modification of the argument below.

Up to changing our choice of unit normal ${\vec{\nu}}$, we can write ${\phi(x) = - \sup\limits_{y\neq x} \frac{\langle\vec{\nu}(x), y\rangle}{1-\langle x,y\rangle}}$. Let us apply the doubling of variables method by considering the function

$\displaystyle Z(x,y) = \phi(x)(1-\langle x, y\rangle) + \langle\vec{\nu}(x), y\rangle\geq 0$

Given a point ${x\in\Sigma}$, we have two possibilities: either ${\phi(x)=\psi(x)}$ or ${\phi(x)>\psi(x)}$.

In the first case (${\phi(x)=\psi(x)}$), since ${\phi\geq\psi}$, one has (from the baby maximum principle) that ${|d\phi|(x) = |d\psi|(x)}$ and ${\Delta(\phi-\psi)(x)\geq 0}$. By plugging this into Simons formula (7), we deduce that

$\displaystyle \Delta\phi(x)\geq \Delta\psi(x) = \frac{|d\psi|(x)^2}{\psi(x)} - 2(\psi(x)^2-1)\psi(x) = \frac{|d\phi|(x)^2}{\phi(x)} - 2(\psi(x)^2-1)\phi(x)$

In the second case ${\phi(x)>\psi(x)}$, we have that there exists ${y\in \Sigma}$, ${y\neq x}$ such that ${Z(x,y)=0}$. Since ${Z\geq 0}$, we get that ${\nabla Z(x,y)=0}$. Geometrically, this condition means that ${\Sigma}$ stays outside the ball ${B:=B\left(x-\frac{1}{\phi(x)}\vec{\nu}(x), \frac{1}{\phi(x)}\right)}$ and ${\Sigma}$ is tangent to ${B}$ at ${x}$ and ${y}$. In particular, this implies that the tangent planes ${T_x\Sigma}$ and ${T_y\Sigma}$ are symmetric with respect to the mediator hyperplane of the segment ${[x,y]}$ between ${x}$ and ${y}$. By exploiting this symmetry, Brendle chose good coordinate systems on ${T_x\Sigma}$ and ${T_y\Sigma}$ leading him to the following inequality

$\displaystyle \begin{array}{rcl} 0&\leq& \sum\left(\frac{\partial^2}{\partial x_i^2} + \frac{\partial^2}{\partial y_i^2} + 2 \frac{\partial^2}{\partial x_i\partial y_i}\right)Z(x,y) \\ &\leq& \left(\Delta\phi(x) -\frac{|d\phi|(x)^2}{\phi(x)} + 2(\psi(x)^2-1)\phi(x)\right)(1-\langle x, y\rangle) \\ &-& C(x,y)(\phi(x)^2-\psi(x)^2)|d\phi|(x)^2 \end{array}$

with ${C(x,y) = (1-\langle x,y\rangle)/4\phi(x)^3}$ after seven pages of calculations in his paper! Since ${C(x,y)>0}$, we have the good sign to conclude from this estimate that

$\displaystyle \Delta\phi(x) -\frac{|d\phi|(x)^2}{\phi(x)} + 2(\psi(x)^2-1)\phi(x)\geq 0$

as it was claimed.

Once we know that ${\phi}$ is a (viscosity) solution of (8), we can use the maximum principles, the inequality ${\phi\geq \psi}$ and the Simons formula (7) for ${\psi}$ to obtain that

$\displaystyle \phi=\kappa\psi$

where ${\kappa\geq 1}$ is a constant.

We claim that this implies that ${\psi}$ is constant, so that the proof of Brendle theorem would be complete (by Lawson’s result quoted right after Simons formula (7)). Indeed, we have again two cases: either ${\kappa=1}$ or ${\kappa>1}$.

In the first situation, since the oscullating balls ${B^{\pm}_{x,R}}$ with ${R=1/\phi(x)}$ have the same principal curvature ${\phi(x)=\psi(x)}$ of ${\Sigma}$ at ${x}$, a third order expansion of (the graph of) ${\Sigma}$ at ${x}$ reveals that ${d\phi(x)=0}$ for all ${x\in\Sigma}$, so that ${\phi=\psi}$ is constant.

In the second situation ${\phi=\kappa\psi}$ with ${\kappa>1}$, we look again at the inequality

$\displaystyle \begin{array}{rcl} 0&\leq& \left(\Delta\phi(x) -\frac{|d\phi|(x)^2}{\phi(x)} + 2(\psi(x)^2-1)\phi(x)\right)(1-\langle x, y\rangle) \\ &-& C(x,y)(\phi(x)^2-\psi(x)^2)|d\phi|(x)^2 \end{array}$

showed above (under the assumption ${\phi(x)>\psi(x)}$, which is our current situation at all ${x\in\Sigma}$ since ${\phi=\kappa\psi}$, ${\kappa>1}$).

Because ${\psi}$ satisfies Simons formula (7) and ${\phi=\kappa\psi}$ with ${\kappa(>1)}$ a constant, we get that the first term of the previous inequality vanishes. In particular, we deduce that

$\displaystyle 0\leq -C(x,y)(\phi(x)^2-\psi(x)^2)|d\phi|(x)^2$

This means that ${|d\phi|(x)^2=0}$ for all ${x\in\Sigma}$ (since ${C(x,y)>0}$ and ${\phi\geq \psi}$), so that ${\psi=\phi/\kappa}$ is constant, as it was claimed.

This completes the proof of Brendle’s theorem and, consequently, the discussion of this post.