Self-Similar Variables and Monotonicity

6A.12 Entropy, Information and Diffusion ended with a template. A monotone quantity along a flow, such as an entropy, has an equality case, and the equality case is a self-similar solution: the Gaussian for the heat equation, a self-shrinker for mean curvature flow, a shrinking soliton for the Ricci flow. So if a limit of rescalings near a singularity makes the monotone quantity constant, the limit is self-similar. That one sentence is the logical core of Perelman's analysis of singularities, and it hides three pieces of mathematics that this chapter makes explicit.

The first is a change of variables in which self-similar solutions stand still. Self-similar solutions are fixed points of the scaling symmetry; after dividing out the scaling, by measuring space in units of $\sqrt{t}$ or $\sqrt{T - t}$ and time logarithmically, they become stationary solutions of a new equation. The second is the discovery that in these variables many flows become gradient flows in a Gaussian-weighted space, so that their energies are automatically monotone: the monotonicity formulas of Huisken, Giga–Kohn and Struwe are nothing but the energy identity of a gradient flow, written in the original variables. The third is the blow-up argument: a bounded monotone quantity has a limit, so its derivative is integrable, so along a sequence the time derivative of the rescaled solution tends to zero, so the limit is stationary in the rescaled variables, that is, self-similar in the original ones.

We do this first for the linear heat equation, where everything is explicit and the self-similar variables turn the heat equation into the Ornstein–Uhlenbeck equation, whose spectrum is the Hermite spectrum; then for the semilinear heat equation of 6A.8 Nonlinear Parabolic Equations, following Yoshikazu Giga and Robert Kohn; then for mean curvature flow, where Huisken's formula of 6A.10 Curve Shortening and the First Geometric Flows reappears as the energy identity of a weighted area; and finally for the Ricci flow, where Perelman's $\mathcal W$-entropy and reduced volume play the same role, in the form in which Morgan and Tian present them.

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

• classify self-similar solutions of the heat equation as forward, backward and translating, and give their geometric-flow analogues;
• transform the heat equation into forward self-similar variables, recognise the Fokker–Planck form with the Gaussian as equilibrium, and prove convergence to the Gaussian at the rate $t^{-1/2}$;
• prove the Gaussian Poincaré inequality with its sharp constant from the Mehler formula, and compute the Hermite spectrum of the Ornstein–Uhlenbeck operator;
• write the semilinear heat equation in backward self-similar variables, prove that it is a gradient flow in $L^2(\rho\,dy)$ with the Giga–Kohn Lyapunov functional, and explain what this says about blow-up profiles;
• derive Struwe-type monotonicity for the heat equation and read Huisken's formula as a weighted energy identity;
• prove monotonicity of Poon's parabolic frequency and explain its relation to unique continuation;
• describe Colding–Minicozzi entropy and White's regularity theorem, and lay out the blow-up argument step by step;
• match each step to its counterpart in Perelman's work and in Morgan–Tian.

13.1Self-similar solutions

The heat equation $u_t = \Delta u$ on $\mathbb{R}^n$ is invariant under the parabolic dilations $u_\lambda(x, t) = u(\lambda x, \lambda^2t)$, and, being linear, under multiplication by constants. A solution is self-similar if dilation maps it to a multiple of itself: $u(\lambda x, \lambda^2t) = \lambda^{-a}u(x, t)$ for all $\lambda > 0$ and some exponent $a$. Setting $\lambda = t^{-1/2}$ shows that such a solution has the form

a fixed profile $V$ that spreads as $\sqrt t$ and changes height as $t^{-a/2}$. The heat kernel is the case $a = n$, $V(y) = (4\pi)^{-n/2}e^{-|y|^2/4}$. Self-similar solutions come in three kinds, according to how they sit in time, and each has counterparts for the geometric flows.

kind	form	heat equation	mean curvature flow	Ricci flow
forward (expanding)	$t^{-a/2}V(x/\sqrt t)$, $t > 0$	heat kernel; data homogeneous of degree $-a$	expanders coming out of cones	expanding solitons; the Gaussian soliton
backward (shrinking)	$(T - t)^{-a/2}V(x/\sqrt{T - t})$, $t < T$	solutions blowing up at time $T$	self-shrinkers: spheres, cylinders	shrinking solitons: spheres, cylinders
translating (steady)	$e^{ct}V(x - c't)$ or $V(x - ct)$	travelling waves (6A.9 Reaction–Diffusion: Fronts, Patterns and Pinching)	the grim reaper (6A.10 Curve Shortening and the First Geometric Flows)	steady solitons: the cigar, Bryant's soliton

Forward self-similar solutions describe how a solution forgets its initial data; backward ones describe how it approaches a singularity; translating ones are eternal and appear as limits of type II singularities (6A.8 Nonlinear Parabolic Equations, 11B.5 Singularities). The Ricci flow column is where the guide is going: shrinking solitons are the models of type I singularities, and Perelman proved that his $\kappa$-solutions look, at very large scales, like shrinking solitons (12B.1 κ-Solutions).

Homogeneous data give self-similar solutions. If $u_0(x) = |x|^{-a}$ with $0 < a < n$, then $u_0(\lambda x) = \lambda^{-a}u_0(x)$; the solution $u = \Phi(\cdot, t)*u_0$ satisfies $u(\lambda x, \lambda^2t) = \lambda^{-a}u(x, t)$ because both sides solve the heat equation with the same initial data, and the solution is unique in the class of 6A.3 The Heat Equation on ℝⁿ (Exercise 13.10). This is the simplest instance of a general principle: a scale-invariant problem has scale-invariant solutions, provided its solutions are unique. The same principle produces Ricci expanders out of Euclidean cones, and it fails exactly when uniqueness fails.

13.2Forward self-similar variables: every solution becomes the Gaussian

To divide out the scaling, measure space in units of $\sqrt t$ and time by its logarithm:

The prefactor $t^{-n/2}$ preserves mass: $\int u(x, t)\,dx = \int v(y, \tau)\,dy$. A short computation (Exercise 13.6) gives

This is a Fokker–Planck equation (6D.3 Stochastic Differential Equations and Feynman–Kac): the heat equation with a drift that pushes mass towards the origin with velocity $-\frac y2$, exactly compensating the spreading. Its stationary solutions with finite mass are the multiples of

because $\nabla G + \frac y2G = 0$. In the new variables the heat kernel is a fixed point, and the question "does every solution look like the heat kernel for large time?" becomes "does every solution of this equation converge to its equilibrium?"

To answer it, write $v = Gf$. Since $\nabla v + \frac y2v = G\nabla f$, the equation becomes

the Ornstein–Uhlenbeck equation. The operator $L$ is symmetric in $L^2(G\,dy)$, by the integration by parts $\int(Lf)h\,G = -\int\nabla f\cdot\nabla h\,G$, and nonpositive. Constants are in its kernel. Convergence to equilibrium is the decay of $f - M$, where $M = \int fG = \int v$ is the conserved mass, and the rate is the spectral gap of $-L$. The gap is the best constant in a Poincaré inequality for the Gaussian measure.

The constant $2$ is the variance of each coordinate under $G$, and the inequality, unlike the Poincaré inequality on a bounded domain (4A.10 Sobolev Spaces), has a constant independent of the dimension. The proof uses the explicit solution of the Ornstein–Uhlenbeck equation, the Mehler formula

which solves $\partial_sP_sf = LP_sf$ with $P_0f = f$ (Exercise 13.7). It says that the Ornstein–Uhlenbeck process started at $y$ is, at time $s$, a Gaussian centred at $e^{-s/2}y$: the drift pulls the starting point in, the noise spreads it out, and as $s \to \infty$ the distribution becomes $G$ whatever the start.

The rate is sharp: a solution whose centre of mass is not at the origin, such as the heat kernel started at a point $x_0 \neq 0$, differs from $M\Phi$ at relative order $\frac{|x_0|}{\sqrt t}$, which is the first Hermite mode $f - M \propto y$ decaying like $e^{-\tau/2}$. Subtracting the right Gaussian (centred at the centre of mass) improves the rate to $t^{-1}$, the next eigenvalue. This is the PDE form of the central limit theorem: the heat equation is the law of Brownian motion (6A.3 The Heat Equation on ℝⁿ), and its convergence to the Gaussian, with the corrections organised by the Hermite spectrum, is the Edgeworth expansion of probability. The relative entropy version, with the log-Sobolev inequality in place of the Poincaré inequality, is the entropy production argument of 6A.12 Entropy, Information and Diffusion.

The Hermite spectrum. The eigenfunctions of $-L$ are the Hermite polynomials adapted to $G$: in one dimension, $H_0 = 1$, $H_1 = y$, $H_2 = y^2 - 2$, $H_3 = y^3 - 6y$, ..., with $-LH_k = \frac k2H_k$ (Exercise 13.8), and in $n$ dimensions products of these, with eigenvalue half the total degree. The eigenvalue $\frac k2$ corresponds, back in the original variables, to a correction decaying like $t^{-k/2}$ relative to the Gaussian. In the backward variables of the next section the same eigenfunctions are the heat polynomials in disguise: $x$, $x^2 + 2t$, $x^3 + 6xt$, ..., the polynomial solutions of the heat equation, written for $t < 0$ as $(-t)^{k/2}H_k(x/\sqrt{-t})$ (Exercise 13.8). Their appearance there is what makes the frequency function work.

13.3Backward self-similar variables and the Giga–Kohn functional

Now consider a solution that becomes singular at time $T$ at the origin, and look at it on the scale of the singularity. For the semilinear equation

of 6A.8 Nonlinear Parabolic Equations, the spatially constant solution $\kappa(T - t)^{-1/(p-1)}$, with $\kappa = (p - 1)^{-1/(p-1)}$, blows up at time $T$ at the ODE rate, and the equation is invariant under $u \mapsto \lambda^{2/(p-1)}u(\lambda x, \lambda^2t)$. Giga and Kohn (1985) introduced the backward self-similar variables

As $t \uparrow T$, $s \to \infty$, and the singularity at the origin is spread out over all of $y$-space: a point at fixed $y$ approaches the origin at the rate $\sqrt{T - t}$. A computation like the forward one (Exercise 13.9) gives

The drift now points outward, $-\frac y2\cdot\nabla w$, undoing the concentration of the singularity, and the constant solution of the ODE becomes the stationary solution $w \equiv \kappa$. The key observation is that the linear part is $\frac1\rho\operatorname{div}(\rho\nabla w)$ with the Gaussian weight

because $\nabla\rho = -\frac y2\rho$. So the equation is the gradient flow, in $L^2(\rho\,dy)$, of the weighted energy below.

Three consequences follow, and they are the template for everything after.

1. Integrability of the time derivative. If $E$ is bounded below along the flow (Giga and Kohn prove this for blowing-up solutions; it is where real work is needed), then $\int_{s_0}^\infty\!\int w_s^2\rho\,dy\,ds \leq E[w(s_0)] - \lim E < \infty$. So $w_s \to 0$ in an averaged sense, and every limit of $w(\cdot, s_k + \cdot)$ along a sequence $s_k \to \infty$ is a stationary solution of the rescaled equation.
2. Stationary means self-similar. A stationary solution $w(y)$ corresponds to $u = (T - t)^{-1/(p-1)}w(x/\sqrt{T - t})$, a backward self-similar solution. So the blow-up of $u$ at the origin is asymptotically self-similar.
3. A Liouville theorem finishes it. Giga and Kohn proved (1985, 1987) that for $1 < p < \frac{n + 2}{n - 2}$ (any $p > 1$ if $n \leq 2$) the only bounded stationary solutions are $0$ and $\pm\kappa$, using a Pohozaev identity (6C.3 Symmetry, Noether’s Theorem and Moving Planes). For positive solutions they also proved the type I bound $u \leq C(T - t)^{-1/(p-1)}$ in the same range (1987), so $w$ is bounded. Hence $w(\cdot, s) \to \kappa$ (at a blow-up point) uniformly on compact sets: in self-similar variables the solution near the singularity is flat, and the singularity looks like the ODE.

The convergence $w \to \kappa$ is the precise form of the slogan of 6A.9 Reaction–Diffusion: Fronts, Patterns and Pinching, "near a singularity reaction wins". It also hides a finer structure: the convergence is only on compact sets of $y$, that is, on regions of size $O(\sqrt{T - t})$ in $x$, and the shape of the singular region is decided by a slowly decaying correction. Jean Bricmont and Antti Kupiainen (1994) and Frank Merle and Hatem Zaag (1997) showed that the generic profile is

a flat top whose width grows like $\sqrt{s} = \sqrt{|\log(T - t)|}$ in self-similar variables. The logarithm comes from the Hermite mode $H_2$, whose eigenvalue for the linearisation at $\kappa$ is zero: a neutral direction, along which the approach to the self-similar solution is only logarithmic. The same mechanism, with the same kind of logarithmic correction, governs the shape of the Ricci flow neckpinch analysed by Angenent and Knopf (11B.6 Rotationally Symmetric Flows and the Standard Solution).

13.4Monotonicity formulas are weighted energy identities

Undo the change of variables in the Giga–Kohn identity and you get a statement about $u$ itself, a monotonicity formula in which the Gaussian weight has become the backward heat kernel centred at the singular point,

which solves the backward heat equation $\partial_t\Phi + \Delta\Phi = 0$ and concentrates at $x_0$ as $t \uparrow T$. The simplest example is the linear heat equation itself.

The quantity $u_t - \frac{(x - x_0)\cdot\nabla u}{2(T - t)}$ is, up to the factor $\frac{1}{T - t}$, the derivative of $u$ along the generator of the parabolic dilations centred at $(x_0, T)$: it vanishes exactly for functions invariant under them, as one checks on $u = V\big((x - x_0)/\sqrt{T - t}\big)$. Every monotonicity formula in this chapter has the form "a scale-invariant weighted energy decreases at the rate of the squared failure of self-similarity". Here is the list.

• Harmonic map heat flow. Michael Struwe (1988) proved the same formula, word for word, for maps $u : \mathbb{R}^n \to N$ into a Riemannian manifold flowing by the harmonic map heat flow (6C.7 Harmonic Maps and Their Heat Flow): $(T - t)\int|\nabla u|^2\Phi$ is nonincreasing, and constant only for backward self-similar maps. Combined with an $\varepsilon$-regularity theorem, it bounds the size of the singular set of the flow.
• Mean curvature flow. Huisken's formula (6A.10 Curve Shortening and the First Geometric Flows) is the same identity for the area of a moving hypersurface $M_t \subset \mathbb{R}^{n+1}$ against the backward heat kernel of $\mathbb{R}^{n+1}$ restricted to $M_t$, normalised by $(4\pi(T - t))^{-n/2}$:

In backward variables, the rescaled flow is the gradient flow of the Gaussian area $F(\Sigma) = (4\pi)^{-n/2}\int_\Sigma e^{-|y|^2/4}d\mu$, and its stationary points, the hypersurfaces with $H = \frac{\langle y, \nu\rangle}{2}$, are the self-shrinkers: minimal surfaces for the conformally flat metric $e^{-|y|^2/2n}\delta$. A sphere of radius $\sqrt{2n}$ and cylinders $S^k(\sqrt{2k})\times\mathbb{R}^{n-k}$ are self-shrinkers (Exercise 13.11).

• Semilinear heat equation. The Giga–Kohn functional, written in the original variables, is a monotone combination of $(T - t)^{(p+1)/(p-1)}\int\big(\frac12|\nabla u|^2 - \frac{|u|^{p+1}}{p+1}\big)\Phi$ and $(T - t)^{2/(p-1)}\int u^2\Phi$.
• Ricci flow. Perelman's reduced volume (12A.5 Reduced Distance and Reduced Volume) is a Gaussian-weighted volume $\int(4\pi\tau)^{-n/2}e^{-\ell}\,dV$, with $\tau$ the backward time from the base point and $\ell$ Perelman's reduced distance playing the role of $\frac{|x - x_0|^2}{4\tau}$. It is nonincreasing in $\tau$, it tends to $1$ as $\tau \to 0$ at a regular base point, and it is constant on an interval only for a gradient shrinking soliton (on flat space, the Gaussian). His $\mathcal W$-entropy (12A.3 The 𝓦-Entropy) is a Gaussian-weighted log-Sobolev quantity, monotone along the flow coupled to the conjugate heat equation, constant exactly on gradient shrinking solitons.

The orientation conventions differ (Perelman's $\tau$ runs backward), and the Ricci flow versions need the weight to evolve with the metric, so it is not a Euclidean Gaussian but the conjugate heat kernel or $e^{-\ell}$; but the structure is identical. Morgan and Tian present the reduced volume first as an integral of a Gaussian-type density along $\mathcal L$-geodesics and prove its monotonicity by a comparison argument (their chapters 6–8), rather than as a gradient flow; the equality case is analysed in the same way, and gives the asymptotic soliton of a $\kappa$-solution (12B.1 κ-Solutions).

13.5Frequency functions and unique continuation

Monotonicity formulas measure more than energy. For harmonic functions, Frederick Almgren (1979) introduced the frequency

proved that it is nondecreasing in $r$, and found that it is constant exactly for homogeneous harmonic polynomials, for which it equals the degree. Nicola Garofalo and Fang-Hua Lin (1986) used it to prove unique continuation: a solution of an elliptic equation that vanishes to infinite order at a point vanishes identically, because the frequency would have to be infinite. Chi-Cheung Poon (1996) found the parabolic version, and in backward self-similar variables it is a two-line computation.

Undoing the variables, $s \to \infty$ is $t \uparrow T$: approaching the point $(0, T)$ of space-time, the frequency decreases to a limit, and the limit is $\frac k2$ for the lowest Hermite mode present in $w$, which is half the parabolic order to which $u$ vanishes at $(0, T)$. (For $u = 1 + (x^2 + 2t)$ with $T = 0$ the limit is $0$, because $u(0, 0) \neq 0$; for $u = x^2 + 2t$ it is $1$.) Here is the idea of the unique continuation argument. Suppose $u$ vanished to infinite order at $(0, T)$. Then every Hermite mode would be absent in the limit, so $N(s) \to \infty$; but $N$ is nonincreasing and finite at any one time, a contradiction unless $w \equiv 0$. This is the mechanism of unique continuation and backward uniqueness for parabolic equations, which Book 6B develops, and of which the Ricci flow has a version: Brett Kotschwar (2010) proved backward uniqueness for the Ricci flow, so a Ricci flow is determined by its metric at any one time, and isometries cannot be created or destroyed by the flow.

13.6The blow-up argument, step by step

Here is the argument in the form in which it recurs, with its four steps named.

1. Monotone quantity. Fix a space-time point $(x_0, T)$, at or near which the solution becomes singular. A scale-invariant quantity $\Theta(t)$, built with the backward heat kernel centred there, is monotone and bounded. So $\Theta(t)$ has a limit $\Theta_0$ as $t \uparrow T$, the density of the flow at $(x_0, T)$, and the defect $\int|\text{failure of self-similarity}|^2\Phi$ is integrable in the logarithmic time $s$.
2. Compactness. Rescale parabolically about $(x_0, T)$ by factors $\lambda_k \to \infty$. The rescaled solutions satisfy uniform estimates (from the equation and from bounds on the monotone quantity), so a subsequence converges, smoothly or weakly, to a limit flow, called a tangent flow.
3. Rigidity. By scale invariance, the monotone quantity of the $k$-th rescaling at time $-1$ equals $\Theta(T - \lambda_k^{-2})$, which tends to $\Theta_0$; the same holds at every negative time. So the limit has constant $\Theta$, and the equality case of the monotonicity formula says that it is self-similar: a self-shrinker, a shrinking soliton, a self-similar harmonic map, the constant $\kappa$.
4. Classification. Self-similar solutions satisfy an elliptic equation (stationary in self-similar variables), and under natural conditions they can be classified. The classification is a Liouville theorem.

For mean curvature flow of mean convex surfaces, step 4 is Huisken's classification (1990, 1993): the only self-shrinkers with $H \geq 0$ and bounded second fundamental form are the round spheres and the generalised cylinders $S^k\times\mathbb{R}^{n-k}$. For the semilinear heat equation it is the Giga–Kohn Liouville theorem. For the Ricci flow in dimension three, Perelman showed that the asymptotic solitons of $\kappa$-solutions are the round sphere and its quotients, the round cylinder $S^2\times\mathbb{R}$, and its $\mathbb{Z}_2$ quotient (12B.1 κ-Solutions); this is the classification that makes the canonical neighbourhood theorem possible (12B.5 The Canonical Neighbourhood Theorem).

Density and regularity. The density $\Theta_0$ also distinguishes regular points from singular ones. For mean curvature flow the density at a point where the flow is smooth is exactly $1$ (the Gaussian area of a plane), and Brian White (2005) proved an $\varepsilon$-regularity theorem: there is $\varepsilon > 0$ such that if the Gaussian density ratios at all nearby points and scales are less than $1 + \varepsilon$, the flow is smooth nearby with curvature bounds. Perelman's pseudolocality theorem (12A.9 Pseudolocality) is the Ricci flow counterpart: a region that is almost Euclidean at some scale, as measured by an isoperimetric inequality and a curvature lower bound, stays regular for a definite time.

Entropy. Tobias Colding and William Minicozzi (2012) took the supremum of the Gaussian area over all centres and scales,

the entropy of a hypersurface. It is invariant under dilations and translations, nonincreasing under mean curvature flow by Huisken's formula, and it is $1$ for a plane, $\sqrt{2\pi/e} \approx 1.52$ for a circle and $\frac4e \approx 1.47$ for a round two-sphere (Exercise 13.11). They proved that spheres, cylinders and planes are the only self-shrinkers that cannot be perturbed to lower entropy, so that a generic flow develops only spherical and cylindrical singularities. Perelman's $\nu$-entropy, the infimum of $\mu(g, \tau)$ over all scales $\tau$, plays the same role for the Ricci flow, and the noncollapsing theorem of 12A.4 κ-Noncollapsing is its first consequence.

13.7History

Self-similar solutions of the heat equation are as old as the heat kernel; the change to logarithmic time and rescaled space was used for the porous medium equation by Barenblatt in the 1950s. Almgren's frequency function appeared in 1979 and Garofalo and Lin's unique continuation theorem in 1986. Giga and Kohn's paper on self-similar blow-up appeared in 1985, with sequels in 1987 and 1989. Struwe's monotonicity formula for the harmonic map flow appeared in 1988, Huisken's for mean curvature flow in 1990, and his classification of mean convex shrinkers in 1990 and 1993. Bricmont and Kupiainen's and Merle and Zaag's work on blow-up profiles dates from 1994 and 1997. Poon's parabolic frequency appeared in 1996. Perelman's entropy and reduced volume appeared in 2002, White's regularity theorem in 2005, Kotschwar's backward uniqueness for the Ricci flow in 2010, and Colding and Minicozzi's entropy in 2012.

13.8Exercises

Solution

By the chain rule, $\partial_t[v(x/\sqrt t, \log t)] = -\frac{x}{2t^{3/2}}\cdot\nabla v + \frac1tv_\tau = \frac1t\big(v_\tau - \frac y2\cdot\nabla v\big)$, so $u_t = t^{-n/2 - 1}\big(-\frac n2v - \frac y2\cdot\nabla v + v_\tau\big)$, while $\Delta_xu = t^{-n/2 - 1}\Delta_yv$. Equating gives the equation. Since $\operatorname{div}(\frac y2v) = \frac n2v + \frac y2\cdot\nabla v$, it is $\operatorname{div}(\nabla v + \frac y2v)$. For $G = (4\pi)^{-n/2}e^{-|y|^2/4}$, $\nabla G = -\frac y2G$, so the flux vanishes. It corresponds to $u = t^{-n/2}(4\pi)^{-n/2}e^{-|x|^2/4t} = \Phi(x, t)$, the heat kernel.

Solution

(a) $\int e^{\xi\cdot z\sqrt{1 - e^{-s}}}G(z)\,dz = e^{(1 - e^{-s})|\xi|^2}$, because $\int e^{\eta\cdot z}G(z)\,dz = e^{|\eta|^2}$ for the Gaussian of variance $2$ per coordinate. So $P_se^{\xi\cdot y} = \exp(e^{-s/2}\xi\cdot y + (1 - e^{-s})|\xi|^2) =: e^{\phi}$. Then $\partial_se^\phi = \big(-\frac12e^{-s/2}\xi\cdot y + e^{-s}|\xi|^2\big)e^\phi$, while $\nabla e^\phi = e^{-s/2}\xi e^\phi$, $\Delta e^\phi = e^{-s}|\xi|^2e^\phi$, so $Le^\phi = \big(e^{-s}|\xi|^2 - \frac12e^{-s/2}\xi\cdot y\big)e^\phi$, the same. At $s = 0$ it is $e^{\xi\cdot y}$. Polynomials are obtained by differentiating in $\xi$ at $\xi = 0$. (b) $\int\!\!\int f(e^{-s/2}y + \sqrt{1 - e^{-s}}z)G(y)G(z)\,dy\,dz$: if $Y$, $Z$ are independent with density $G$, then $e^{-s/2}Y + \sqrt{1 - e^{-s}}Z$ again has density $G$ (a Gaussian with covariance $(e^{-s} + 1 - e^{-s})\cdot2I$), so the integral is $\int fG$.

Solution

(a) $-L(y^k) = -k(k - 1)y^{k-2} + \frac k2y^k$. (b) $-L(y^2 + c) = -2 + y^2$, which equals $(y^2 + c)$ times $1$ when $c = -2$: $H_2 = y^2 - 2$, eigenvalue $1 = \frac22$. $-L(y^3 + cy) = -6y + \frac32y^3 + \frac c2y$, and setting this equal to $\frac32(y^3 + cy)$ gives $-6 + \frac c2 = \frac{3c}{2}$, $c = -6$: $H_3 = y^3 - 6y$. (c) $\partial_t(x^2 + 2t) = 2 = \partial_x^2(x^2)$ and $\partial_t(x^3 + 6xt) = 6x = \partial_x^2(x^3)$. With $T = 0$ and $t < 0$, $y = x/\sqrt{-t}$: $x^2 + 2t = (-t)(y^2 - 2) = (-t)H_2(y)$ and $x^3 + 6xt = (-t)^{3/2}(y^3 - 6y) = (-t)^{3/2}H_3(y)$. With $s = -\log(-t)$, $w = u$ is $e^{-s}H_2$ and $e^{-3s/2}H_3$: eigenfunctions decaying at the rates $\frac k2$, as Proposition 13.5 says.

Solution

(a) Write $\theta = T - t$, so $\frac{ds}{dt} = \frac1\theta$ and $\frac{\partial y}{\partial t} = \frac{y}{2\theta}$. With $u = \theta^{-1/(p-1)}w$: $u_t = \theta^{-1/(p-1) - 1}\big(\frac{w}{p - 1} + \frac y2\cdot\nabla w + w_s\big)$, $\Delta u = \theta^{-1/(p-1) - 1}\Delta w$, and $|u|^{p-1}u = \theta^{-p/(p-1)}|w|^{p-1}w = \theta^{-1/(p-1) - 1}|w|^{p-1}w$. Equating gives the equation. (b) A constant $w$ satisfies $-\frac{w}{p - 1} + |w|^{p-1}w = 0$, so $w = 0$ or $|w|^{p-1} = \frac{1}{p - 1}$. The solution $u = \pm\kappa(T - t)^{-1/(p-1)}$ is the ODE solution blowing up at $T$. (c) The derivative of $-\frac{w}{p - 1} + |w|^{p-1}w$ at $\kappa$ is $-\frac{1}{p - 1} + p\kappa^{p-1} = -\frac{1}{p - 1} + \frac{p}{p - 1} = 1$. So the linearisation is $L + 1$, with eigenvalues $1 - \frac k2$ on $H_k$: positive (unstable) for $k = 0, 1$, zero (neutral) for $k = 2$, negative for $k \geq 3$. The neutral mode $H_2$ is responsible for the logarithmic profile of Bricmont–Kupiainen and Merle–Zaag.

Solution

(a) $u(\lambda x, \lambda^2t) = \int\Phi(\lambda x - z, \lambda^2t)|z|^{-a}dz$; substitute $z = \lambda z'$: $dz = \lambda^ndz'$, $\Phi(\lambda(x - z'), \lambda^2t) = \lambda^{-n}\Phi(x - z', t)$ and $|z|^{-a} = \lambda^{-a}|z'|^{-a}$, giving $\lambda^{-a}u(x, t)$. (b) Take $\lambda = t^{-1/2}$: $u(x/\sqrt t, 1) = t^{a/2}u(x, t)$. As $|y| \to \infty$, $V(y) = \int\Phi(y - z, 1)|z|^{-a}dz \approx |y|^{-a}$, since the kernel has unit mass and concentrates within distance $O(1)$ of $y$, where $|z|^{-a} \approx |y|^{-a}$. So far from the origin the solution still looks like its initial data, and the self-similar profile interpolates between a smooth cap and the cone-like tail: the same picture as a Ricci expander coming out of a cone.

Solution

(a) For the sphere, $H = \frac nR$ and $\langle y, \nu\rangle = R$ on it, so $\frac nR = \frac R2$, $R^2 = 2n$. For the cylinder, the $\mathbb{R}^{n-k}$ factor contributes no curvature and is orthogonal to $\nu$, so $H = \frac kR$ and $\langle y, \nu\rangle = R$, giving $R^2 = 2k$. (b) $n = 1$: $(4\pi)^{-1/2}\cdot2\pi\sqrt2\cdot e^{-1/2} = \frac{2\pi\sqrt2}{2\sqrt\pi\sqrt e} = \sqrt{\frac{2\pi}{e}} \approx 1.520$. $n = 2$: $(4\pi)^{-1}\cdot4\pi\cdot4\cdot e^{-1} = \frac4e \approx 1.472$. (c) For $\mathbb{R}^n \subset \mathbb{R}^{n+1}$, $(4\pi)^{-n/2}\int_{\mathbb{R}^n}e^{-|y|^2/4}dy = 1$. These are the entropies: for shrinkers, Colding and Minicozzi showed the supremum in $\lambda$ is attained at $x_0 = 0$, $t_0 = 1$.

Solution

Since $\Delta u = 0$, $\operatorname{div}(u\nabla u) = |\nabla u|^2$, so by the divergence theorem $\int_{B_r}|\nabla u|^2 = \int_{\partial B_r}u\,\partial_ru$. By homogeneity, $x\cdot\nabla u = ku$, so on $\partial B_r$, $\partial_ru = \frac{x}{r}\cdot\nabla u = \frac kru$. Then $r\int_{B_r}|\nabla u|^2 = r\cdot\frac kr\int_{\partial B_r}u^2$, and $N(r) = k$. The frequency counts the degree, the order of vanishing of $u$ at the origin, just as Poon's frequency counts the order of a caloric polynomial.