Fourier Methods and Scale Space

6A.3 The Heat Equation on ℝⁿ solved the heat equation on $\mathbb{R}^n$ by one formula, convolution with the heat kernel. This chapter looks at the same solutions from the other side, the side of frequencies. On the Fourier side the heat equation is diagonal: each frequency $\xi$ decays by itself, at the rate $4\pi^2|\xi|^2$, independently of all the others. Almost every property of the heat flow becomes a one-line statement about the multiplier $e^{-4\pi^2t|\xi|^2}$: smoothing is the decay of high frequencies, the approach to equilibrium on a compact space is the decay of the lowest nonzero frequency, backward ill-posedness is the explosion of high frequencies, and the parabolic scaling $x \mapsto \lambda x$, $t \mapsto \lambda^2t$ is the statement that $t|\xi|^2$ is the only combination that matters.

The second half of the chapter is about a use of the heat equation that most people meet every day without knowing it. Blurring a digital photograph by a Gaussian is running the heat equation for a time proportional to the square of the blur radius, and in the 1980s computer vision built on this a whole theory, scale space, in which an image is replaced by the family of all its heat-flow smoothings. The maximum principle of 6A.5 Maximum Principles becomes a design requirement ("blurring must not create new features"), the Gaussian turns out to be the only kernel that satisfies it, and the failures of linear smoothing at edges led to nonlinear diffusions that are close relatives of the geometric flows of 6A.10 Curve Shortening and the First Geometric Flows, including flows by curvature. The chapter ends by reading the Ricci flow in the same Fourier terms: its linearisation is a heat equation for the metric, apart from the directions that diffeomorphisms move, which is the frequency-side picture of DeTurck's trick.

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

• solve the heat equation on $\mathbb{R}^n$ and on the torus with the Fourier transform and Fourier series, and read off smoothing, decay and uniqueness from the multiplier $e^{-4\pi^2t|\xi|^2}$;
• prove the Sobolev smoothing estimate $\|u(t)\|_{\dot H^s} \leq C_st^{-(s - r)/2}\|g\|_{\dot H^r}$ and explain the exponent;
• write the heat kernel of the circle and of the torus in two ways, as a Fourier series and as a periodised Gaussian, and explain why they agree;
• solve the heat equation on an interval with Dirichlet or Neumann conditions by sine and cosine series, and compute the rate of approach to equilibrium;
• describe fractional and anomalous diffusion as Fourier multipliers, and where they occur;
• explain linear scale space, the causality axiom, why the Gaussian is forced, and the Laplacian-of-Gaussian and difference-of-Gaussians detectors;
• describe the Perona–Malik equation, total variation flow and curvature-based image smoothing, and their relation to the maximum principle and to geometric flows;
• explain, on the Fourier side, why the Ricci flow is a heat equation for the metric except in the gauge directions.

4.1Frequencies decay independently

With the normalisation $\hat g(\xi) = \int g(x)e^{-2\pi ix\cdot\xi}\,dx$ of 4A.6 The Fourier Transform, differentiation becomes multiplication: $\widehat{\partial_jg}(\xi) = 2\pi i\xi_j\hat g(\xi)$ and $\widehat{\Delta g}(\xi) = -4\pi^2|\xi|^2\hat g(\xi)$. Transforming the heat equation $u_t = \Delta u$ in $x$ gives, for each $\xi$, the ordinary differential equation $\partial_t\hat u(\xi, t) = -4\pi^2|\xi|^2\hat u(\xi, t)$, whose solution is

6A.3 The Heat Equation on ℝⁿ inverted this to find the heat kernel: the inverse transform of a Gaussian is a Gaussian, and multiplication on the Fourier side is convolution on the space side. Here we stay on the Fourier side and ask what the multiplier says.

The heat semigroup is a Fourier multiplier. Write $e^{t\Delta}g$ for the solution at time $t$. Then $\widehat{e^{t\Delta}g} = m_t\hat g$ with $m_t(\xi) = e^{-4\pi^2t|\xi|^2}$. Three properties of $m_t$ translate into three properties of the flow.

• $0 < m_t \leq 1$, so by Plancherel $\|e^{t\Delta}g\|_{L^2} \leq \|g\|_{L^2}$: the flow does not increase the $L^2$ norm. It is a contraction.
• $m_tm_s = m_{t+s}$: the semigroup law $e^{t\Delta}e^{s\Delta} = e^{(t+s)\Delta}$, which on the space side is $\Phi(\cdot, t)*\Phi(\cdot, s) = \Phi(\cdot, t + s)$.
• $m_t(\xi) \to 1$ as $t \to 0$ for each $\xi$, and dominated convergence gives $e^{t\Delta}g \to g$ in $L^2$: the flow starts where it should.

Energy and uniqueness. By Plancherel, $\int|u(x, t)|^2\,dx = \int e^{-8\pi^2t|\xi|^2}|\hat g(\xi)|^2\,d\xi$, which is nonincreasing in $t$, and

the energy identity of 6A.3 The Heat Equation on ℝⁿ in Fourier form. A solution in $L^2$ with $g = 0$ has $\hat u = 0$ for all $t$; this is uniqueness among $L^2$ solutions, without any growth condition at infinity beyond square integrability. (Tychonoff's nonunique solution of 6A.3 The Heat Equation on ℝⁿ is not square integrable at positive times: it grows faster than any Gaussian, and its Fourier transform does not exist as a function.)

4.2Smoothing is the decay of high frequencies

On the Fourier side, smoothness is decay: a function has $s$ derivatives in $L^2$ exactly when $|\xi|^s\hat g \in L^2$ (4A.10 Sobolev Spaces). Define the homogeneous Sobolev seminorms

so that $\|g\|_{\dot H^1} = \|\nabla g\|_{L^2}$ and $\|g\|_{\dot H^2} = \|\Delta g\|_{L^2}$.

The exponent $\frac{s - r}{2}$ is the parabolic counting that recurs throughout the book: one space derivative costs a factor $t^{-1/2}$, the square root of the time, because time scales like length squared. 6A.3 The Heat Equation on ℝⁿ proved the pointwise version, $|\nabla^ku| \leq Ct^{-k/2}\sup|g|$, by differentiating the kernel. Shi's estimates for the Ricci flow (11A.5 Shi's Derivative Estimates) are the same statement for the curvature tensor, $|\nabla^k\operatorname{Rm}| \leq C_kKt^{-k/2}$, proved by the maximum principle because the Ricci flow has no Fourier transform; but the exponent is the one the multiplier gives.

Why backward is hopeless. Running the equation backward multiplies frequency $\xi$ by $e^{+4\pi^2t|\xi|^2}$. Unless $\hat g$ decays faster than every Gaussian, the result is not in $L^2$ for any $t > 0$; and even when it is, an error of size $\varepsilon$ at frequency $|\xi| = N$ grows to $\varepsilon e^{4\pi^2tN^2}$. This is the frequency-side form of the ill-posedness proved in 6A.1 What a PDE Is, and the reason the conjugate heat equation in Perelman's work is always solved backward in time, from a final condition, where for it the problem is well posed (9B.7 The Heat Equation on a Manifold).

4.3The circle and the torus: two formulas for one kernel

On the circle $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, 2B.7 Fourier Series and the First Heat Equation solved the heat equation by Fourier series, as Fourier himself did:

Exchanging the sum and the integral writes $u = p_t*g$ with the heat kernel of the circle

On the other hand, the circle is the line modulo translations by integers, and a solution of the heat equation on $\mathbb{R}$ that is periodic is a solution on $\mathbb{T}$. Periodising the Gaussian gives a second formula:

The two agree by the Poisson summation formula $\sum_mf(x + m) = \sum_k\hat f(k)e^{2\pi ikx}$ for a Schwartz function $f$ (4A.6 The Fourier Transform), applied to the Gaussian $f = \Phi(\cdot, t)$, whose transform is $e^{-4\pi^2t\xi^2}$.

The two formulas are good at opposite ends of time. For small $t$ the Gaussian sum is dominated by its $m = 0$ term: the kernel is nearly the Euclidean one, because heat has not yet had time to feel that the space closes up. For large $t$ the Fourier sum is dominated by $k = 0$ and $k = \pm1$:

so heat equilibrates to its mean at the rate $e^{-4\pi^2t}$, the rate of the first nonzero eigenvalue $\lambda_1 = 4\pi^2$ of $-\Delta$ on the circle. The same duality between short-time geometry and long-time spectrum governs the heat kernel of any closed manifold (9B.7 The Heat Equation on a Manifold): the small-time expansion of the heat trace sees volume and curvature, the large-time behaviour sees the spectral gap. The identity between the two formulas, the Jacobi theta identity, is the simplest instance of a deep link between heat kernels and number theory.

On the torus $\mathbb{T}^n = \mathbb{R}^n/\mathbb{Z}^n$ everything factorises: $p_t(x) = \prod_jp_t^{(1)}(x_j)$, and the spectral gap is still $4\pi^2$, attained by the frequencies $\pm e_j$. Flat tori of other shapes, $\mathbb{R}^n/\Lambda$ for a lattice $\Lambda$, have kernels $\sum_{\ell \in \Lambda}\Phi(x + \ell, t)$, and their spectra are $\{4\pi^2|\xi|^2 : \xi \in \Lambda^*\}$, with $\Lambda^*$ the dual lattice. Two flat tori with the same spectrum are said to be isospectral; Milnor found in 1964 two sixteen-dimensional flat tori that are isospectral but not isometric, the first answer, in the negative, to the question "can one hear the shape of a drum?" in a closed setting (9B.7 The Heat Equation on a Manifold).

4.4Intervals: sines, cosines and boundary conditions

On an interval $[0, L]$ the boundary conditions select which frequencies are allowed.

• Dirichlet ($u = 0$ at both ends, the ends held at temperature zero). The eigenfunctions of $-\partial_x^2$ are $\sin\frac{k\pi x}{L}$, $k \geq 1$, with eigenvalues $\frac{k^2\pi^2}{L^2}$, and

Every solution decays to zero, at the rate $e^{-\pi^2t/L^2}$ of the first mode.

• Neumann ($u_x = 0$ at both ends, insulated ends, 6A.1 What a PDE Is). The eigenfunctions are $\cos\frac{k\pi x}{L}$, $k \geq 0$, and the $k = 0$ mode is the constant, which does not decay: $u \to \frac1L\int_0^Lg$, the mean, conserved because no heat leaves. The approach is at the rate $e^{-\pi^2t/L^2}$ again.

The method is the odd and even reflection of the circle: extending $g$ oddly about $0$ and periodically with period $2L$ produces a function on a circle of length $2L$ whose series contains only sines, and the flow preserves oddness, so it vanishes at $0$ and $L$ for all time. In both cases the time to equilibrate is of order $\frac{L^2}{\pi^2}$: doubling the length of a rod quadruples the time it takes to even out. This is Einstein's law $\langle x^2\rangle \sim t$ of 6A.3 The Heat Equation on ℝⁿ again, read off from the lowest eigenvalue.

For a general bounded domain the eigenfunctions of the Dirichlet Laplacian are no longer explicit, but they exist and form an orthonormal basis of $L^2$ (the spectral theorem for the compact inverse, 4A.8 Compact Operators and Spectra, 6B.15 Spectral Theory and Heat Trace Asymptotics), and the solution is the same kind of series. The first eigenfunction is positive (6A.6 Weak Solutions and Elliptic Regularity), and solutions become proportional to it for large time: a hot object of any shape cools, after a while, in the shape of its first eigenfunction.

4.5Fractional and anomalous diffusion

Replacing the multiplier $e^{-4\pi^2t|\xi|^2}$ by $e^{-t(2\pi|\xi|)^\alpha}$, $0 < \alpha < 2$, defines the fractional heat equation $u_t = -(-\Delta)^{\alpha/2}u$. Its kernel is the density of a symmetric $\alpha$-stable Lévy process, a random motion with occasional long jumps; it is positive (by a theorem of Bochner, since $e^{-|\xi|^\alpha}$ is positive definite for $\alpha \leq 2$), but has heavy tails, decaying like $|x|^{-n-\alpha}$ instead of like a Gaussian. Its scaling is $x \sim t^{1/\alpha}$ instead of $x \sim t^{1/2}$: superdiffusion. The case $\alpha = 1$ is the Poisson kernel of the half-space, $\frac{c\,t}{(t^2 + |x|^2)^{(n+1)/2}}$: the harmonic extension of $g$ to the upper half-space, evaluated at height $t$ (6A.2 Harmonic Functions).

Anomalous diffusion is observed: tracer particles in turbulent flows, the foraging paths of some animals, and the spread of some contaminants in heterogeneous porous media have been modelled with Lévy-type processes and fractional equations, though how well such models fit is debated case by case. None of this enters the Ricci flow, whose diffusion is genuinely second order; but it shows that what makes the heat equation special is the exponent $2$, which is the only exponent compatible with locality (a differential operator) and with positivity of the kernel at the same time.

4.6Scale space: the heat equation in computer vision

A digital image is a function $g$ on a grid, sampled from an intensity on a plane region. To find its features (edges, blobs, corners) at all sizes at once, Andrew Witkin (1983) and Jan Koenderink (1984) proposed replacing $g$ by the whole family of its smoothings

the scale-space representation: $L$ solves the heat equation with $L(\cdot, 0) = g$, and the scale parameter $t$ corresponds to a blur radius $\sigma = \sqrt{2t}$, the standard deviation of the Gaussian $\Phi(\cdot, t)$. Fine structure lives at small $t$, coarse structure at large $t$, and moving from fine to coarse is running the heat equation.

Why the Gaussian. Koenderink asked what a smoothing should satisfy to deserve the name, and listed requirements: it should be linear, shift-invariant and rotation-invariant; it should form a semigroup in the scale parameter (smoothing by $t$ then by $s$ is smoothing by $t + s$); and it should satisfy causality: going to a coarser scale must never create new structure. For level sets, causality means that a new local maximum cannot appear as $t$ increases, and Koenderink observed that this is exactly what a parabolic maximum principle provides: if $L$ satisfies $L_t = \Delta L$, then at a spatial maximum $\Delta L \leq 0$, so $L_t \leq 0$, and maxima can only decrease (6A.5 Maximum Principles). Requiring causality for every image forces the generator of the semigroup to be a second-order elliptic operator, and the symmetry requirements make it the Laplacian: the heat equation is the only linear scale space. Babaud, Witkin, Baudin and Duda (1986) proved the one-dimensional form precisely: among smoothing kernels, the Gaussian is the only one for which the number of zero crossings of the second derivative (inflection points of the signal) never increases with scale.

There is a subtlety that the maximum principle makes precise. In one dimension the heat flow cannot create new local extrema of the signal (a consequence of the maximum principle applied to $L_x$, which also solves the heat equation, combined with the fact that the number of sign changes of a solution of the one-dimensional heat equation does not increase, a theorem going back to Sturm). In two dimensions new local extrema can appear under Gaussian smoothing: Lifshitz and Pizer (1990) gave examples. Causality survives in the weaker form that matters: no new level curve can appear from nothing, because a closed level curve enclosing a new maximum would need $L_t > 0$ at that maximum.

Detectors from derivatives. Features are defined through derivatives of $L$, and the scale is chosen by normalising them. David Marr and Ellen Hildreth (1980) located edges at the zero crossings of $\Delta L = \Delta(\Phi*g) = (\Delta\Phi)*g$, the Laplacian of Gaussian, motivated in part by the centre–surround receptive fields of retinal ganglion cells. Since $\partial_t\Phi = \Delta\Phi$, the Laplacian of Gaussian is approximately the difference of two Gaussians at nearby scales,

which is cheap to compute: the heat equation itself tells you that blurring twice and subtracting gives the Laplacian. David Lowe's SIFT detector (2004), used for years in image matching, panorama stitching and robot navigation, finds keypoints as extrema of the difference of Gaussians across both space and scale. Blob detection with the scale-normalised Laplacian $t\Delta L$ (Lindeberg) selects, for each blob, the scale at which $t\Delta L$ is extremal, and that scale is proportional to the square of the blob's size: parabolic scaling, again.

4.7Nonlinear diffusion: smoothing that respects edges

Linear scale space blurs edges as much as noise, because the Laplacian is isotropic: it does not know which way an edge runs. In 1990 Pietro Perona and Jitendra Malik proposed letting the diffusion coefficient depend on the image itself,

so that diffusion is strong where the gradient is small (inside regions, removing noise) and weak where it is large (across edges, preserving them). Writing the equation in a frame aligned with the gradient shows what it does: along the level curves the diffusion coefficient is $c(|\nabla u|) > 0$, but across them it is $\frac{d}{ds}\big(sc(s)\big)$, which is negative for $s > K$. So the equation is a forward heat equation along edges and a backward heat equation across steep ones. That sharpens edges, which is what was wanted, but it also makes the problem ill-posed in the sense of 6A.1 What a PDE Is: small perturbations can grow, and the continuous equation has no well-behaved solutions in general (it develops "staircasing" under discretisation). Catté, Lions, Morel and Coll (1992) regularised it by computing the coefficient from a Gaussian-smoothed gradient, which restores well-posedness.

Two other lines of work keep the maximum principle intact.

• Total variation flow. Leonid Rudin, Stanley Osher and Emad Fatemi (1992) denoised images by minimising the total variation $\int|\nabla u|$ plus a fidelity term. Its gradient flow, $u_t = \operatorname{div}\frac{\nabla u}{|\nabla u|}$, is degenerate but not backward: it is the gradient flow of a convex functional (6A.11 Calculus of Variations and Gradient Flows), preserves the maximum principle, and allows jumps in $u$, which is why it keeps edges sharp. The quantity $\operatorname{div}\frac{\nabla u}{|\nabla u|}$ is the curvature of the level sets of $u$.
• Curvature motion of level sets. Luis Alvarez, Pierre-Louis Lions and Jean-Michel Morel (1992), and Alvarez, Guichard, Lions and Morel's axiomatic study (1993), showed that the image smoothings that are contrast-invariant (unchanged by relabelling the grey levels monotonically) and satisfy the scale-space axioms are generated by equations of the form $u_t = |\nabla u|F(\kappa, t)$, where $\kappa$ is the curvature of the level curves. The simplest, $u_t = |\nabla u|\kappa$, moves every level curve by the curve shortening flow of 6A.10 Curve Shortening and the First Geometric Flows; the affine-invariant one, $u_t = |\nabla u|\kappa^{1/3}$, is the affine curve shortening of Sapiro and Tannenbaum. Such equations are fully nonlinear and degenerate; their solutions are defined as viscosity solutions (6C.9 Hamilton–Jacobi Equations and Viscosity Solutions).

So the practical demand of smoothing images without destroying them leads, by a short chain of reasoning, to the same geometric flows that this book develops as mathematics: the heat equation, curvature flows of curves, and the maximum principle as the organising idea.

4.8The Ricci flow, seen from the Fourier side

The Ricci flow has no Fourier transform, but its linearisation at a flat metric does, and that linearisation explains the role of every tool of Book 11A. Linearise $\partial_tg = -2\operatorname{Ric}(g)$ at the Euclidean metric: write $g = \delta + h$ with $h$ small, and keep terms linear in $h$. The variation formula of 11A.2 How Curvature Evolves gives

Transform in $x$: with $\hat h(\xi)$ a symmetric matrix for each frequency,

The matrix in brackets is the symbol of the linearised Ricci operator (11A.4 Short-Time Existence and Uniqueness, 6C.12 Quasilinear Parabolic Systems). On most symmetric matrices it acts like $|\xi|^2$, so those components decay like the heat equation. But on the matrices $\hat h = \xi v^T + v\xi^T$, for any vector $v$, the bracket vanishes (Exercise 4.6): these components do not decay at all. They are the infinitesimal changes of coordinates, the Fourier transforms of the Lie derivatives $\partial_jX_k + \partial_kX_j$, and the Ricci flow cannot see them because a diffeomorphism changes nothing geometric. DeTurck's trick (11A.4 Short-Time Existence and Uniqueness) adds a Lie-derivative term $\mathcal L_Wg$, with $W$ built from first derivatives of $g$, whose symbol is exactly what is missing in these directions: the symbol of the modified flow is $|\xi|^2$ times the identity on all symmetric matrices, and the modified flow is a genuine heat equation for the metric. Short-time existence of the Ricci flow is, at the level of frequencies, the statement that after fixing coordinates every component of the metric diffuses.

The other Fourier fact that runs through Perelman's work is the Gaussian itself. The Gaussian is the fixed point of the Fourier transform (up to scaling), the heat kernel, the extremal function of the log-Sobolev inequality (6A.12 Entropy, Information and Diffusion), and the density $(4\pi\tau)^{-n/2}e^{-|x|^2/4\tau}$ at which Perelman's entropy $\mathcal W$ vanishes on flat space (12A.3 The 𝓦-Entropy). That one function plays all these roles is not a coincidence: they are all expressions of the fact that $e^{-4\pi^2t|\xi|^2}$ is both the heat multiplier and a Gaussian.

4.9History

Joseph Fourier solved the heat equation by trigonometric series in his memoir of 1807 and in Théorie analytique de la chaleur (1822). Poisson's summation formula dates from the 1820s and Jacobi's theta identity from 1828. Bochner characterised positive-definite functions in 1932, and Lévy introduced stable laws in the 1920s. Marr and Hildreth's theory of edge detection appeared in 1980, Witkin's scale-space filtering in 1983 and Koenderink's "The structure of images" in 1984; Babaud, Witkin, Baudin and Duda proved the uniqueness of the Gaussian kernel in 1986. Perona and Malik introduced anisotropic diffusion in 1990, Rudin, Osher and Fatemi total variation denoising in 1992, and Alvarez, Lions and Morel curvature-based smoothing in 1992. Lowe's SIFT descriptor was published in 2004. Milnor's isospectral tori appeared in 1964.

4.10Exercises

Solution

By Parseval on the torus, $\|u(t)\|^2 = \sum_{k \neq 0}e^{-8\pi^2t|k|^2}|\hat g(k)|^2$, since $\hat g(0) = 0$ (mean zero). For $k \in \mathbb{Z}^n\setminus\{0\}$, $|k|^2 \geq 1$, so each term is at most $e^{-8\pi^2t}|\hat g(k)|^2$, and $\|u(t)\|^2 \leq e^{-8\pi^2t}\|g\|^2$. Equality for some $t > 0$ forces $\hat g(k) = 0$ whenever $|k|^2 > 1$, that is, $g$ is supported on the frequencies $\pm e_j$, the modes $e^{\pm2\pi ix_j}$; conversely those modes decay exactly at the rate $e^{-4\pi^2t}$.

Solution

If $\hat g$ is supported in a thin shell $\rho \leq |\xi| \leq \rho + \delta$, then on that shell $(2\pi|\xi|)^{2s}e^{-8\pi^2t|\xi|^2} \approx (2\pi\rho)^{2(s-r)}e^{-8\pi^2t\rho^2}(2\pi|\xi|)^{2r}$, so $\|e^{t\Delta}g\|_{\dot H^s}^2 \approx (2\pi\rho)^{2(s-r)}e^{-8\pi^2t\rho^2}\|g\|_{\dot H^r}^2$. With $y = 4\pi^2\rho^2t$ this is $\big(y^{(s-r)/2}e^{-y}\big)^2t^{-(s-r)}\|g\|^2_{\dot H^r}$, and choosing $y = \frac{s - r}{2}$, where $y^{(s-r)/2}e^{-y}$ attains its maximum $C_{s,r}$, gives the bound up to an error that vanishes as $\delta \to 0$.

Solution

Gaussian form at $x = 0$: $p_t(0) = (4\pi t)^{-1/2}\sum_me^{-m^2/4t} = (4\pi t)^{-1/2}(1 + 2e^{-1/4t} + \ldots)$. Fourier form: $p_t(0) = \sum_ke^{-4\pi^2k^2t} = 1 + 2e^{-4\pi^2t} + 2e^{-16\pi^2t} + \ldots$. Equating the two forms with $4\pi t = s$: $\sum_ke^{-\pi k^2s} = s^{-1/2}\sum_me^{-\pi m^2/s}$, Jacobi's identity for the theta function $\theta(s) = \sum e^{-\pi k^2s}$, which says $\theta(s) = s^{-1/2}\theta(1/s)$.

Solution

The cosine coefficients of $x$ on $[0, 1]$: $a_0 = \int_0^1x\,dx = \frac12$, and for $k \geq 1$, $a_k = 2\int_0^1x\cos k\pi x\,dx = \frac{2((-1)^k - 1)}{k^2\pi^2}$, which is $-\frac{4}{k^2\pi^2}$ for odd $k$ and $0$ for even $k$. So $u(x, t) = \frac12 - \sum_{k\text{ odd}}\frac{4}{k^2\pi^2}e^{-k^2\pi^2t}\cos k\pi x$. As $t \to \infty$, $u \to \frac12$, the mean, at the rate $e^{-\pi^2t}$ of the $k = 1$ mode. $\int_0^1u\,dx = \frac12$ for all $t$, since each cosine with $k \geq 1$ integrates to zero; directly, $\frac{d}{dt}\int u = \int u_{xx} = u_x\big|_0^1 = 0$.

Solution

For $\hat h = \xi v^T + v\xi^T$: $\operatorname{tr}\hat h = 2\xi\cdot v$ and $\hat h\xi = |\xi|^2v + (v\cdot\xi)\xi$. Then $\xi(\hat h\xi)^T + (\hat h\xi)\xi^T = |\xi|^2(\xi v^T + v\xi^T) + 2(v\cdot\xi)\xi\xi^T$, which cancels $|\xi|^2\hat h + 2(\xi\cdot v)\xi\xi^T$ exactly: the symbol is zero. These are the Fourier transforms of $\mathcal L_Xg$, infinitesimal diffeomorphisms, which the Ricci flow does not damp. If $\hat h$ is trace-free with $\hat h\xi = 0$ (transverse, trace-free: the "physical" part), the last three terms vanish and the symbol is $|\xi|^2\hat h$: these components evolve by the heat equation. (For gravitational waves, the analogous transverse trace-free part is what propagates.)

Solution

(a) By Taylor's theorem in $t$ and $\partial_t\Phi = \Delta\Phi$: $\Phi(t + \delta) - \Phi(t) = \delta\partial_t\Phi(t) + \frac{\delta^2}{2}\partial_t^2\Phi(t + \theta\delta) = \delta\Delta\Phi(t) + O(\delta^2)$, uniformly in $x$ for $t$ bounded away from $0$. In two dimensions $\Phi = \frac{1}{4\pi t}e^{-r^2/4t}$ and $\Delta\Phi = \partial_t\Phi = \frac{1}{4\pi t}\big(\frac{r^2}{4t^2} - \frac1t\big)e^{-r^2/4t}$: negative in the centre, positive outside, the "Mexican hat". (b) $L(\cdot, t) = \Phi(\cdot, a)*\Phi(\cdot, t) = \Phi(\cdot, a + t)$ by the semigroup law, so $\Delta L(0, t) = \partial_s\Phi(0, s)|_{s=a+t} = -\frac{1}{4\pi(a + t)^2}$, and $t\Delta L(0, t) = -\frac{t}{4\pi(a + t)^2}$. Its derivative in $t$ vanishes where $(a + t) - 2t = 0$, at $t = a$, where $|t\Delta L|$ is largest.