Eigenvalue Problems and Perelman’s λ

6B.15 Spectral Theory and Heat Trace Asymptotics studied the spectrum of the Laplacian as a whole, through the heat trace $\sum e^{-\lambda_kt}$, and found that it knows the volume, the boundary and the total scalar curvature. This chapter studies the eigenvalues one at a time, above all the first, which governs almost everything that happens for long times: the rate at which heat equilibrates on a bounded region (6A.4 Fourier Methods and Scale Space), whether a population survives in a habitat (6A.9 Reaction–Diffusion: Fronts, Patterns and Pinching), the ground state energy of a quantum particle, the lowest note of a drum, the load at which a column buckles. Its eigenfunction, the ground state, has a property no other eigenfunction has: it is positive. And the first eigenvalue is characterised by a minimisation, the Rayleigh quotient, so it can be estimated from above by any test function and from below by inequalities of geometric content.

The chapter ends where Perelman's first preprint begins. Perelman defined

the first eigenvalue of the Schrödinger operator $-4\Delta + R$, whose potential is the scalar curvature, and proved that it increases along the Ricci flow (12A.2 Ricci Flow as a Gradient Flow). Everything Perelman uses about $\lambda$ before that monotonicity, that the infimum is attained, by a unique positive smooth function, that $\lambda$ depends smoothly on the metric, that it is bounded between the minimum and the average of the scalar curvature, is the ground-state theory of this chapter, which Perelman states in a line ("Clearly, $\lambda(g_{ij})$ is just the lowest eigenvalue of the operator $-4\triangle + R$") and Topping and Kleiner–Lott spell out.

Until Book 8A defines manifolds, "closed manifold" can be read as the torus $\mathbb{T}^n = \mathbb{R}^n/\mathbb{Z}^n$ with a periodic metric $g_{ij}(x)$ (6B.10 Harmonic Coordinates and the Regularity of Metrics): a compact space without boundary, with the Laplace–Beltrami operator $\Delta_g$ and volume $dV = \sqrt{|g|}\,dx$. Every argument below uses only integration by parts, the Rellich compactness theorem and elliptic regularity, all of which hold there.

By the end of this chapter you will be able to:

• characterise eigenvalues by Rayleigh quotients and the min–max principle, and prove domain monotonicity and the Dirichlet–Neumann ordering;
• prove that the first eigenvalue of a Schrödinger operator $-\Delta_g + V$ on a closed manifold is attained, simple, with a positive smooth eigenfunction, and that every other eigenfunction changes sign;
• use the ground-state substitution to prove Hardy's inequality with its sharp constant, and compute the hydrogen ground state;
• state Faber–Krahn and prove Cheeger's inequality $\lambda_1 \geq h^2/4$;
• explain spectral clustering, the Fiedler vector and the Perron–Frobenius analogy behind PageRank;
• define Perelman's $\lambda$, derive its Euler–Lagrange equation in the variables $w$ and $f$, prove $\min R \leq \lambda \leq \bar R$ and the scaling law, compute it on model spaces, and relate its sign to the Yamabe problem;
• derive the first variation of a simple eigenvalue (Hellmann–Feynman) and explain why $\lambda$ is a smooth function of the metric.

16.1Rayleigh quotients and the min–max principle

Let $H$ be a self-adjoint operator, bounded below, with compact resolvent, on a Hilbert space: for example $-\Delta$ with Dirichlet conditions on a bounded domain (6B.15 Spectral Theory and Heat Trace Asymptotics), $-\Delta$ with Neumann conditions, or $-\Delta_g + V$ on a closed manifold with $V$ bounded. By the spectral theorem for compact operators (4A.8 Compact Operators and Spectra) its spectrum is a sequence of eigenvalues $\lambda_1 \leq \lambda_2 \leq \cdots \to \infty$, repeated with multiplicity, with an orthonormal basis of eigenfunctions $\phi_k$. Write $Q(u) = \langle Hu, u\rangle$ for the quadratic form, extended to its form domain (for $-\Delta_g + V$, $Q(u) = \int(|\nabla u|^2 + Vu^2)\,dV$ on $H^1$).

Three consequences follow at once, and they are why the variational characterisation is so useful.

• Upper bounds from test functions. Any trial function $u$ gives $\lambda_1 \leq Q(u)/\|u\|^2$. This is Rayleigh's principle, which Lord Rayleigh used in The Theory of Sound (1877) to estimate the frequencies of strings, membranes and organ pipes. Walther Ritz (1909) minimised over finite-dimensional families of trial functions, and the Rayleigh–Ritz method is the ancestor of the finite element method (6D.10 Monte Carlo, Fourier and Finite Element Methods): engineers compute the natural frequencies of buildings and bridges by min–max over spaces of piecewise polynomials, and the results are automatically upper bounds.
• Domain monotonicity. If $\Omega \subset \Omega'$, every function in $H^1_0(\Omega)$ extends by zero to one in $H^1_0(\Omega')$ with the same Rayleigh quotient, so $\lambda_k^D(\Omega') \leq \lambda_k^D(\Omega)$: a larger drum sounds lower.
• Dirichlet above Neumann. $H^1_0(\Omega) \subset H^1(\Omega)$, and both problems have the quadratic form $\int|\nabla u|^2$, so $\lambda_k^N(\Omega) \leq \lambda_k^D(\Omega)$: clamping the boundary raises every frequency. (The first Neumann eigenvalue is $0$, with constant eigenfunction.)

16.2The ground state is positive and simple

The same theorem holds for the Dirichlet problem on a connected bounded domain, with the same proof (the Hopf lemma of 6A.5 Maximum Principles adds that $\partial_\nu\phi_1 < 0$ on a smooth boundary). Its discrete analogue is the Perron–Frobenius theorem: a matrix with positive entries has a simple largest eigenvalue with a positive eigenvector. In both cases positivity comes from the same source, the fact that the operator (or its resolvent, or its heat semigroup) preserves positivity: the heat kernel is positive (6A.3 The Heat Equation on ℝⁿ), so $e^{-tH}$ maps positive functions to positive functions, and the ground state is the eigenvector of $e^{-tH}$ with the largest eigenvalue $e^{-t\lambda_1}$.

16.3The ground-state substitution

A positive eigenfunction does more than exist: it can be factored out. Let $\phi > 0$ be smooth with $-\Delta\phi + V\phi = \mu\phi$ on an open set $U \subseteq \mathbb{R}^n$ (or on a manifold). For $u = \phi w$ with $w$ smooth and compactly supported in $U$,

(Exercise 16.5). The quadratic form of $-\Delta + V - \mu$, in the variable $w = u/\phi$, is a pure Dirichlet energy with weight $\phi^2$. Two consequences:

Positivity of the form. If there is a positive solution of $-\Delta\phi + V\phi = \mu\phi$ on $U$, then $-\Delta + V \geq \mu$ on functions supported in $U$. Positive (super)solutions are certificates of lower bounds for the spectrum (a principle going back to Jacobi's study of conjugate points, and known as the Allegretto–Piepenbrink theorem in spectral theory).

Hardy's inequality. On $\mathbb{R}^n\setminus\{0\}$, $n \geq 3$, the function $\phi = |x|^{-(n-2)/2}$ satisfies $-\Delta\phi = \frac{(n - 2)^2}{4}\frac{\phi}{|x|^2}$ (Exercise 16.6). Taking $V = -\frac{(n - 2)^2}{4|x|^2}$ and $\mu = 0$ gives

with the best constant (it is approached by $u \approx \phi$ cut off near $0$ and $\infty$, for which $\int\phi^2|\nabla w|^2$ is relatively small). Hardy's inequality is the uncertainty principle of quantum mechanics in a sharp form: a particle cannot be both localised near the origin and have small kinetic energy. It is also scale invariant, like Sobolev's inequality (4A.11 Sobolev Embeddings and Critical Exponents), and both reappear in the analysis of singularities.

The hydrogen atom. In units where the Schrödinger operator of the electron is $-\Delta - \frac{2}{|x|}$ on $\mathbb{R}^3$ (energies in Rydbergs, lengths in Bohr radii), $\phi = e^{-|x|}$ is a positive eigenfunction with eigenvalue $-1$ (Exercise 16.7). By the positivity criterion it is the ground state: the binding energy of hydrogen is one Rydberg, about $13.6$ electronvolts, and the ground state wave function decays exponentially with the Bohr radius as its length scale. The excited states, with energies $-\frac{1}{n^2}$, have eigenfunctions that change sign, as Theorem 16.2 requires.

16.4Geometry in the first eigenvalue

The first Dirichlet eigenvalue of a domain measures its size and its shape. Two theorems make this precise, one sharp and one robust.

Rayleigh conjectured this in The Theory of Sound (1877) on the basis of computations, and Georg Faber (1923) and Edgar Krahn (1925) proved it. The proof replaces the first eigenfunction $u$ by its symmetric decreasing rearrangement $u^*$, the radial decreasing function on $\Omega^*$ whose superlevel sets $\{u^* > s\}$ are balls with the same volumes as $\{u > s\}$. Rearrangement preserves $\int u^2$, and the Pólya–Szegő inequality $\int|\nabla u^*|^2 \leq \int|\nabla u|^2$, which follows from the coarea formula and the isoperimetric inequality applied to each level set, shows that it does not increase the Dirichlet energy. So $u^*$ is a test function on the ball with smaller Rayleigh quotient. The ball sounds lowest because it is the most isoperimetric shape. For a disc of radius $R$, $\lambda_1 = j_{0,1}^2/R^2$ with $j_{0,1} \approx 2.405$ the first zero of the Bessel function $J_0$.

The isoperimetric inequality enters Faber–Krahn through every level set. Jeff Cheeger (1970) found a robust version that needs only an isoperimetric constant. For a domain $\Omega$, let

the infimum over smooth open sets $A$ with closure in $\Omega$: the cheapest way, in boundary area per unit volume, to cut a piece out of $\Omega$.

The inequality is sharp in order of magnitude, not in constant, and it has a converse of sorts: Peter Buser (1982) proved that on closed manifolds with Ricci curvature bounded below, $\lambda_1$ (now the first nonzero eigenvalue, with $h$ defined by cutting the manifold into two pieces) is bounded above by a constant times $h + h^2$. A manifold with a thin neck has small $h$, because cutting the neck is cheap, and therefore small $\lambda_1$; the first eigenfunction is then nearly constant on each side of the neck, with opposite signs, and changes sign in the neck. The connection between small eigenvalues and necks will be met again: a Ricci flow developing a neckpinch between two large regions has a first nonzero eigenvalue that tends to zero, by Buser's inequality, because cutting the neck becomes cheap.

On manifolds. Lower bounds for the first nonzero eigenvalue of a closed manifold come from curvature. If $\operatorname{Ric} \geq (n - 1)g$, then $\lambda_1 \geq n$ (Lichnerowicz, 1958, proved in 9A.6 The Laplacian and the Bochner Formula by the Bochner formula), with equality only on the round sphere (Obata, 1962); if $\operatorname{Ric} \geq 0$ and the diameter is $d$, then $\lambda_1 \geq \frac{\pi^2}{d^2}$ (Zhong and Yang, 1984, sharpening $\frac{\pi^2}{2d^2}$ of Li and Yau, 1980). These are the estimates that a Ricci-flow argument controlling curvature can turn into spectral control.

16.5Perelman's $\lambda$

Let $(M, g)$ be a closed manifold with scalar curvature $R$. Perelman's

is the first eigenvalue of $-4\Delta + R = 4\big(-\Delta + \frac R4\big)$, a Schrödinger operator with potential $\frac R4$, scaled by $4$. By Theorem 16.2 the infimum is attained by a unique positive smooth $w$ with $\int w^2 = 1$, solving

Perelman writes $w = e^{-f/2}$, which is possible precisely because $w > 0$. Then $4|\nabla w|^2 = |\nabla f|^2e^{-f}$ and $w^2 = e^{-f}$ (12A.2 Ricci Flow as a Gradient Flow), so

the minimum of his energy functional $\mathcal F$, and the minimiser $f$ satisfies

(Exercise 16.8). Perelman remarks in his §1 that $\mathcal F$ had appeared in string theory as the low-energy effective action, with $f$ the dilaton. What makes $\lambda$ useful is that every property needed later is a ground-state property:

• It is attained, by a unique minimiser. Uniqueness is simplicity of the ground state, and it is what makes $f$ a well-defined function of $g$: there is no choice to be made.
• It is a smooth function of the metric. A simple eigenvalue of an operator depending smoothly on parameters depends smoothly on them, with a smooth eigenfunction (Kato's perturbation theory; Exercise 16.11 derives the derivative, assuming differentiability). So $\lambda(g(t))$ is a differentiable function of $t$ along a smooth flow, and its derivative can be computed by differentiating the Rayleigh quotient at the minimiser alone: the change in $w$ does not contribute, because $w$ is a critical point. This is how Perelman computes $\frac{d}{dt}\lambda = 2\int|\operatorname{Ric} + \nabla^2f|^2e^{-f}\,dV \geq 0$ along the Ricci flow (12A.2 Ricci Flow as a Gradient Flow); Kleiner and Lott discuss the smooth dependence of the minimiser in their §7.
• It is trapped by the scalar curvature. Since $4|\nabla w|^2 \geq 0$, $\lambda \geq \min R$. Testing with the constant $w = \operatorname{Vol}^{-1/2}$ gives $\lambda \leq \bar R$, the average of $R$. So

with equality on either side exactly when $R$ is constant (Exercise 16.9).

• It scales like curvature. For a constant $c > 0$, $\lambda(cg) = c^{-1}\lambda(g)$, because $R_{cg} = c^{-1}R_g$, $|\nabla w|_{cg}^2 = c^{-1}|\nabla w|_g^2$, and the normalisation $\int w^2\,dV_{cg} = 1$ rescales $w$ by a constant (Exercise 16.10). So $\lambda$ is not scale invariant, which is why it can rule out steady breathers but not shrinking ones; for those Perelman needed the scale-invariant entropy $\mu$ (12A.3 The 𝓦-Entropy).

Model values. On a manifold of constant scalar curvature, the constant function is the ground state and $\lambda = R$: for the unit round $S^n$, $\lambda = n(n - 1)$; for a flat torus, $\lambda = 0$; for a compact hyperbolic manifold, $\lambda = -n(n - 1)$; for the round cylinder $S^2\times S^1$ with unit sphere factor, $\lambda = 2$. On a metric with a thin neck the ground state of $-4\Delta + R$ concentrates where $R$ is smallest, which is not in the neck (where $R$ is large and positive), and $\lambda$ sees the regions of least scalar curvature.

The sign of $\lambda$ and the Yamabe problem. In dimension $n \geq 3$ the conformal Laplacian $-\frac{4(n - 1)}{n - 2}\Delta + R$ governs whether a metric is conformal to one of positive scalar curvature: its first eigenvalue has the sign of the Yamabe invariant of the conformal class (Kazdan and Warner, 1975), and if it is positive, the conformal factor $u^{4/(n-2)}$ built from its ground state $u$ gives a metric $u^{4/(n-2)}g$ of positive scalar curvature. Since $\frac{4(n - 1)}{n - 2} > 4$, the conformal Laplacian's form dominates Perelman's: if $\lambda(g) > 0$, then the conformal class of $g$ contains a metric of positive scalar curvature (Exercise 16.12). Perelman's monotonicity of $\lambda$ therefore says that along the Ricci flow the metric, if ever "spectrally positive", stays so; the obstructions to positive scalar curvature studied in 9B.6 Scalar Curvature and Topology obstruct $\lambda > 0$.

16.6History

Rayleigh's Theory of Sound (1877) introduced the variational principle for the lowest frequency and conjectured that the disc minimises it. Ritz's method dates from 1909. Perron (1907) and Frobenius (1912) proved the positivity theorem for matrices. Courant formulated the min–max principle around 1920 (Fischer had the matrix case in 1905, and Weyl used related ideas in 1912). Faber and Krahn proved Rayleigh's conjecture in 1923–25, and Pólya and Szegő's book on isoperimetric inequalities in mathematical physics appeared in 1951. Hardy's inequality dates from the 1920s. Lichnerowicz's eigenvalue estimate appeared in 1958 and Obata's rigidity theorem in 1962. Cheeger proved his inequality in 1970 and Buser its converse in 1982; Fiedler introduced algebraic connectivity in 1973, and discrete Cheeger inequalities were proved by Dodziuk (1984) and Alon and Milman (1985). Kazdan and Warner's work on scalar curvature and the conformal Laplacian dates from 1975. Li and Yau's eigenvalue bound appeared in 1980 and Zhong and Yang's sharp version in 1984. PageRank was published in 1998 and normalized cuts in 2000. Perelman's $\lambda$ appeared in his first preprint, in November 2002.

Book 6C turns to nonlinear equations, starting with the regularity of minimisers (6C.1 Regularity of Minimisers and Constraints).

16.7Exercises

Solution

$\nabla u = w\nabla\phi + \phi\nabla w$, so $|\nabla u|^2 = w^2|\nabla\phi|^2 + \phi\nabla\phi\cdot\nabla(w^2) + \phi^2|\nabla w|^2$. Integrating the middle term by parts: $\int\phi\nabla\phi\cdot\nabla(w^2) = -\int w^2\operatorname{div}(\phi\nabla\phi) = -\int w^2(|\nabla\phi|^2 + \phi\Delta\phi)$. So $\int|\nabla u|^2 = \int\phi^2|\nabla w|^2 - \int w^2\phi\Delta\phi = \int\phi^2|\nabla w|^2 + \int w^2\phi(\mu - V)\phi$, using $-\Delta\phi = (\mu - V)\phi$. Since $w^2\phi^2 = u^2$, this is $\int\phi^2|\nabla w|^2 + \int(\mu - V)u^2$, which rearranges to the identity.

Solution

(a) For a radial function $f(r)$, $\Delta f = f'' + \frac{n - 1}{r}f'$. With $f = r^{-a}$: $f'' = a(a + 1)r^{-a-2}$ and $\frac{n - 1}{r}f' = -a(n - 1)r^{-a-2}$, summing to $a(a + 2 - n)r^{-a-2}$. (b) With $a = \frac{n - 2}{2}$, $a + 2 - n = -a$, so $\Delta\phi = -a^2\phi/|x|^2$. Then $\phi$ solves $-\Delta\phi + V\phi = 0$ with $V = -\frac{a^2}{|x|^2}$, and the substitution gives $\int|\nabla u|^2 - a^2\int\frac{u^2}{|x|^2} = \int\phi^2|\nabla w|^2 \geq 0$ for $u$ supported away from $0$. (c) For $n = 2$, $a = 0$ and the constant is $0$: there is no Hardy inequality in the plane, reflecting the fact that $\log|x|$ is harmonic and the plane is "critical" (Brownian motion in the plane is recurrent).

Solution

(a) In three dimensions, $\Delta f = f'' + \frac2rf'$; for $f = e^{-r}$ this is $e^{-r} - \frac2re^{-r}$, so $-\Delta\phi - \frac2r\phi = -e^{-r} + \frac2re^{-r} - \frac2re^{-r} = -\phi$. (b) The substitution with $V = -\frac2r$, $\mu = -1$ gives $\int(|\nabla u|^2 - \frac2ru^2) + \int u^2 = \int\phi^2|\nabla w|^2 \geq 0$. (The functions supported away from the origin are dense in the form domain, so this bounds the whole spectrum below by $-1$, and $e^{-r}$, which is in the form domain, attains it.) (c) With $f = (2 - r)e^{-r/2}$: $f' = (-2 + \frac r2)e^{-r/2}$, $f'' = (\frac32 - \frac r4)e^{-r/2}$, so $\Delta f = (\frac32 - \frac r4 - \frac4r + 1)e^{-r/2} = (\frac52 - \frac r4 - \frac4r)e^{-r/2}$. Then $-\Delta f - \frac2rf = (-\frac52 + \frac r4 + \frac4r - \frac4r + 2)e^{-r/2} = (-\frac12 + \frac r4)e^{-r/2} = -\frac14f$. It vanishes at $r = 2$, as an excited state must.

Solution

$\nabla w = e^{-f/2}\cdot(-\frac12\nabla f)$. Then $\Delta w = \operatorname{div}(-\frac12w\nabla f) = -\frac12\nabla w\cdot\nabla f - \frac12w\Delta f = \frac14w|\nabla f|^2 - \frac12w\Delta f$. So $-4\Delta w + Rw = (2\Delta f - |\nabla f|^2 + R)w$, and dividing by $w > 0$ gives the equation.

Solution

For $\int w^2 = 1$: $\int(4|\nabla w|^2 + Rw^2) \geq \int Rw^2 \geq \min R$, so $\lambda \geq \min R$. The constant $w = \operatorname{Vol}^{-1/2}$ gives $\lambda \leq \frac{1}{\operatorname{Vol}}\int R = \bar R$. If $\lambda = \min R$, the minimiser has $\int|\nabla w|^2 = 0$, so $w$ is constant, and the equation $Rw = \lambda w$ gives $R \equiv \lambda$. If $\lambda = \bar R$, the constant is a minimiser, hence (by simplicity) the ground state, and the equation $-4\Delta w + Rw = \lambda w$ with $w$ constant gives $R \equiv \lambda$.

Solution

If $\int w^2\,dV_g = 1$, then $\tilde w = c^{-n/4}w$ has $\int\tilde w^2\,dV_{cg} = 1$, and $\int(4|\nabla\tilde w|_{cg}^2 + R_{cg}\tilde w^2)\,dV_{cg} = c^{-n/2}\cdot c^{-1}\cdot c^{n/2}\int(4|\nabla w|_g^2 + R_gw^2)\,dV_g$. The correspondence $w \leftrightarrow \tilde w$ is a bijection of the constraint sets, so $\lambda(cg) = c^{-1}\lambda(g)$. Since $\operatorname{Vol}(cg)^{2/n} = c\operatorname{Vol}(g)^{2/n}$, the product is invariant; it is the quantity Perelman uses for expanding breathers (12A.2 Ricci Flow as a Gradient Flow).

Solution

Define $G(s, t) = Q_s(w(t)) - \lambda(t_0)\|w(t)\|_s^2$. Then $\lambda(t) - \lambda(t_0) = G(t, t) + \lambda(t_0)(\|w(t)\|_t^2 - 1) = G(t, t)$, since $\|w(t)\|_t = 1$. Differentiate at $t_0$: $\lambda'(t_0) = \partial_sG + \partial_tG$. The second term is the derivative of $Q_{t_0}(w(t)) - \lambda(t_0)\|w(t)\|_{t_0}^2$, which is $2\langle(H(t_0) - \lambda(t_0))w(t_0), w'(t_0)\rangle_{t_0} = 0$. The first is the derivative in $s$ of $Q_s(w_0) - \lambda(t_0)\|w_0\|_s^2$ with $w_0 = w(t_0)$ fixed, which equals the derivative of the Rayleigh quotient $Q_s(w_0)/\|w_0\|_s^2$ at $s = t_0$ (since $\|w_0\|_{t_0} = 1$ and $Q_{t_0}(w_0) = \lambda(t_0)$).

Solution

(a) $a_n - 4 = \frac{4(n - 1) - 4(n - 2)}{n - 2} = \frac{4}{n - 2} > 0$, so $a_n|\nabla w|^2 \geq 4|\nabla w|^2$ pointwise and the inequality of forms follows. Taking the infimum over $\int w^2 = 1$, $\lambda_L \geq \lambda(g)$; in particular $\lambda(g) > 0$ implies $\lambda_L > 0$. (b) $Lu = \lambda_Lu$, so $R_{\tilde g} = u^{-(n+2)/(n-2)}\lambda_Lu = \lambda_Lu^{-4/(n-2)} > 0$, since $u > 0$. The positivity of the ground state is essential: it is what makes $u^{4/(n-2)}$ a metric.