Geometric analysis · The Lumenary

MotivationH

Equations that find the best shape

Dip a wire loop into soapy water and a film forms that, to the eye, has the least area the loop allows. Bend the loop and the film changes; pull two rings apart and the film between them narrows and suddenly bursts into two flat discs. The Belgian physicist Joseph Plateau studied such films experimentally in the nineteenth century, and the mathematical question — does every closed curve in space bound a surface of least area, and what does it look like? — became the Plateau problem.

The film is governed by a nonlinear partial differential equation: its mean curvature vanishes. This is the prototype of geometric analysis, where nonlinear PDE are used to construct geometric objects, and geometric ideas are used to understand the PDE. Its triumphs include:

the solution of the Plateau problem (Douglas and Radó, around 1930);
the positive mass theorem of general relativity, proved by Schoen and Yau (1979) with minimal surfaces;
Yau's solution of the Calabi conjecture (1978), producing the Calabi–Yau metrics of string theory;
Ricci flow, introduced by Richard Hamilton (1982), with which Grigori Perelman proved Thurston's geometrisation conjecture and the Poincaré conjecture (2002–2003).

A unifying theme: many geometric objects are critical points of a natural functional — area for surfaces, energy for maps, total scalar curvature for metrics — and their gradient flows are nonlinear heat equations: mean curvature flow, the harmonic map heat flow, Ricci flow. Heat flows smooth things out, but nonlinear ones can also form singularities in finite time. Understanding the singularities is often where the geometry lies.

IntuitionH

Critical points and heat flows

Think of the area of a surface with fixed boundary as a function on an infinite-dimensional space of surfaces. A minimal surface is a critical point: every small deformation fixing the boundary changes the area to second order only. Not every critical point is a minimum. The two rings make this visible: for rings close together there are two catenoids spanning them, both critical, and only the fatter one is a local minimum; the thinner one is a saddle point — push it and it collapses. Past a critical distance no catenoid exists, and the film jumps to the two discs.

Gradient flows follow the steepest descent of the functional. For curves in the plane, the gradient of length is curvature, and the curve-shortening flow moves each point with velocity equal to its curvature vector: bumps flatten, dents fill in, and a closed curve shrinks to a round point. For metrics, Hamilton's Ricci flow $\partial_t g = -2\,\mathrm{Ric}(g)$ is (in suitable coordinates) a nonlinear heat equation for the metric itself: positively curved regions shrink, negatively curved ones expand, and the curvature tends to even out. On a round sphere the flow simply shrinks the sphere to a point; on a dumbbell, the neck can pinch off first.

DefinitionsH

Definitions

Definition— mean curvature and minimal surface

For an oriented surface $\Sigma \subset \R^3$ with unit normal $\nu$ , the principal curvatures $\kappa_1, \kappa_2$ are the eigenvalues of the shape operator $-d\nu$ , and the mean curvature is $H = \kappa_1 + \kappa_2$ (some authors divide by 2). $\Sigma$ is minimal if $H \equiv 0$ . Equivalently, for every compactly supported normal variation $\Sigma_t = \{x + t\varphi(x)\nu(x)\}$ ,

\frac{d}{dt}\Big|_{t=0} \mathrm{Area}(\Sigma_t) = -\int_\Sigma H \varphi \, dA = 0.

A minimal surface is stable if moreover the second variation is non-negative for all such $\varphi$ .

Definition— minimal surface equation

A graph $z = u(x, y)$ over a domain $\Omega \subset \R^2$ has area $\int_\Omega \sqrt{1 + |\nabla u|^2}\,dx\,dy$ ; it is minimal if and only if

\operatorname{div}\!\left( \frac{\nabla u}{\sqrt{1 + |\nabla u|^2}} \right) = 0 \iff (1 + u_y^2)\, u_{xx} - 2 u_x u_y\, u_{xy} + (1 + u_x^2)\, u_{yy} = 0.

Definition— harmonic map

For a smooth map $f : (M, g) \to (N, h)$ of Riemannian manifolds, $M$ compact, the energy is $E(f) = \tfrac12 \int_M |df|^2 \, dV_g$ . Critical points of $E$ are harmonic maps; they are the solutions of $\tau(f) = \operatorname{tr}_g \nabla df = 0$ , the tension field equation. Examples: geodesics (maps from an interval or circle), harmonic functions ( $N = \R$ ), isometric minimal immersions, and holomorphic maps between Kähler manifolds.

Definition— Ricci flow, mean curvature flow, Yamabe problem

A family of Riemannian metrics $g(t)$ on a manifold $M$ is a Ricci flow if $\partial_t g = -2\,\mathrm{Ric}(g)$ . A family of hypersurfaces $F(\cdot, t) : \Sigma \to \R^{n+1}$ is a mean curvature flow if $\partial_t F = -H \nu$ (for curves in the plane: the curve-shortening flow, $\partial_t \gamma = \kappa N$ ). The Yamabe problem asks for a metric of constant scalar curvature in a given conformal class $[g] = \{e^{2w} g\}$ on a compact manifold of dimension $n \ge 3$ .

Worked examplesH

Examples

Example— Classical minimal surfaces

The plane, and pieces of it.
The catenoid, obtained by rotating the catenary $y = a\cosh(x/a)$ about the $x$ -axis (Euler, 1744): together with the plane it is the only minimal surface of revolution.
The helicoid $(s\cos t, s\sin t, ct)$ (Meusnier, 1776), the only ruled minimal surface besides the plane (Catalan, 1842). It is locally isometric to the catenoid, and there is a one-parameter family of minimal surfaces bending one into the other.
Scherk's surface $z = \log\cos y - \log\cos x$ over the square $|x|, |y| < \pi/2$ (1834), a doubly periodic surface when continued by reflection.
Enneper's surface, a complete minimal surface with self-intersections, from the Weierstrass–Enneper representation: every simply connected minimal surface in $\R^3$ is $\mathrm{Re}\int (\tfrac12 f(1 - g^2), \tfrac{i}{2} f(1 + g^2), fg)\,dz$ for a holomorphic $f$ and meromorphic $g$ (with $fg^2$ holomorphic).

Non-example: a hemisphere of radius $r$ has $H = 2/r \ne 0$ ; it is a critical point of area only among surfaces enclosing a fixed volume (a soap bubble, of constant mean curvature).

Example— Ricci flow on spheres, cylinders and hyperbolic manifolds

On a round sphere $S^n$ of radius $r$ , $\mathrm{Ric} = (n-1)\, g_{S^n}$ where $g_{S^n}$ is the unit round metric, independent of $r$ . Seeking $g(t) = \rho(t)\, g_{S^n}$ gives $\rho' = -2(n-1)$ , so $\rho(t) = r_0^2 - 2(n-1)t$ : the sphere shrinks homothetically and disappears at $T = r_0^2/(2(n-1))$ . On a hyperbolic manifold ( $\mathrm{Ric} = -(n-1)g_{\mathbb{H}^n}$ ) the same computation gives $\rho(t) = r_0^2 + 2(n-1)t$ : it expands forever. On a cylinder $S^{n-1} \times \R$ the sphere factor shrinks and the line factor stays: this is the model of a neck pinch, the singularity Perelman cut out by surgery.

Example— Curve-shortening flow of a circle

A circle of radius $r$ has curvature $1/r$ , so $r' = -1/r$ and $r(t)^2 = r_0^2 - 2t$ : it shrinks to a point at $T = r_0^2/2$ . Its area $\pi r^2$ decreases at rate exactly $2\pi$ . This is a general fact: for any embedded closed curve, $\frac{dA}{dt} = -\oint \kappa\, ds = -2\pi$ by the theorem of turning tangents (Hopf's Umlaufsatz).

ProofH

Central theorems

Theorem— Theorem 1 (the minimal surface equation; Lagrange 1762)

A $C^2$ function $u$ on a domain $\Omega \subset \R^2$ is a critical point of $A(u) = \int_\Omega \sqrt{1 + |\nabla u|^2}$ among functions with the same boundary values if and only if it satisfies the minimal surface equation.

Proof

Let $W = \sqrt{1 + |\nabla u|^2}$ and $\varphi \in C_c^\infty(\Omega)$ . Differentiating under the integral and integrating by parts,

\frac{d}{d\varepsilon}\Big|_{\varepsilon = 0} A(u + \varepsilon\varphi) = \int_\Omega \frac{\nabla u \cdot \nabla \varphi}{W} = -\int_\Omega \varphi \,\operatorname{div}\!\Big(\frac{\nabla u}{W}\Big).

This vanishes for all $\varphi$ exactly when $\operatorname{div}(\nabla u / W) = 0$ , by the fundamental lemma of the calculus of variations. Expanding, $\partial_x(u_x / W) = (u_{xx} W^2 - u_x(u_x u_{xx} + u_y u_{xy}))/W^3$ and similarly for $y$ ; adding and multiplying by $W^3$ gives $(1 + u_y^2) u_{xx} - 2u_xu_yu_{xy} + (1 + u_x^2)u_{yy} = 0$ . (Geometrically, $\operatorname{div}(\nabla u/W)$ is the mean curvature of the graph with respect to the upward normal.)

∎

Theorem— Proposition 2 (minimal surfaces of revolution)

The surface obtained by rotating the graph of a positive $C^2$ function $y(x)$ about the $x$ -axis is minimal if and only if $y\, y'' = 1 + y'^2$ , and the solutions are the catenaries $y = a\cosh((x - b)/a)$ , $a > 0$ .

Proof

The area is $2\pi\int L\,dx$ with $L = yW$ , $W = \sqrt{1 + y'^2}$ . The Euler–Lagrange equation $\frac{d}{dx}(\partial L/\partial y') = \partial L/\partial y$ reads $\frac{d}{dx}(yy'/W) = W$ , that is $(y'^2 + yy'')/W - yy'^2y''/W^3 = W$ ; multiplying by $W^3$ and using $W^2 - y'^2 = 1$ gives $y'^2W^2 + yy'' = W^4$ , so $yy'' = W^2(W^2 - y'^2) = 1 + y'^2$ . Since $L$ does not depend on $x$ , the Beltrami quantity $L - y'\,\partial L/\partial y' = y/W$ is constant along solutions, say $a > 0$ . The catenary $y = a\cosh(x/a)$ satisfies $y' = \sinh(x/a)$ , $y'' = \cosh(x/a)/a$ and $yy'' = \cosh^2(x/a) = 1 + y'^2$ , with $y/W = a$ (both identities are checked in Lean below). Finally, any initial data $y(x_0) > 0$ , $y'(x_0)$ is matched by a catenary $a\cosh((x - b)/a)$ — take $a = y(x_0)/\sqrt{1 + y'(x_0)^2}$ and $b$ with $\sinh((x_0 - b)/a) = y'(x_0)$ — so by uniqueness for the ODE every solution is a catenary.

∎

Scherk's surface: with $u = \log\cos y - \log\cos x$ we have $u_x = \tan x$ , $u_y = -\tan y$ , $u_{xx} = \sec^2 x$ , $u_{yy} = -\sec^2 y$ , $u_{xy} = 0$ , and the left side of the minimal surface equation is $\sec^2 y \sec^2 x - \sec^2 x \sec^2 y = 0$ (also in Lean).

Theorem— Theorem 3 (minimal graphs minimise area: a calibration argument)

Let $u \in C^2(\overline\Omega)$ solve the minimal surface equation on a bounded domain $\Omega \subset \R^2$ , with graph $G$ . If $S$ is any compact oriented surface in the cylinder $\overline\Omega \times \R$ with $\partial S = \partial G$ , then $\mathrm{Area}(S) \ge \mathrm{Area}(G)$ .

Proof

On $\Omega \times \R$ define the unit vector field $\nu(x, y, z) = (-u_x, -u_y, 1)/W$ , the upward normal of $G$ translated vertically (it does not depend on $z$ ). Its divergence is $-\operatorname{div}(\nabla u / W) + \partial_z(1/W) = 0$ by the minimal surface equation. Let $\omega$ be the 2-form $\omega(X, Y) = \det(\nu, X, Y)$ , the interior product of $\nu$ with the volume form; then $d\omega = (\operatorname{div}\nu)\, dV = 0$ . For any oriented orthonormal pair $X, Y$ , $\omega(X, Y) = \nu \cdot (X \times Y) \le 1$ , with equality when $X \times Y = \nu$ . Hence $\int_S \omega \le \mathrm{Area}(S)$ , while $\int_G \omega = \mathrm{Area}(G)$ because $\nu$ is the normal of $G$ . Since $\partial S = \partial G$ , the cycle $S - G$ bounds a 3-chain $R$ in the cylinder $\overline\Omega \times \R$ (whose second homology vanishes, as $\Omega$ is a planar domain), and Stokes' theorem gives $\int_S \omega - \int_G \omega = \int_R d\omega = 0$ . Therefore $\mathrm{Area}(S) \ge \int_S \omega = \int_G \omega = \mathrm{Area}(G)$ .

∎

The form $\omega$ is a calibration (Harvey–Lawson, 1982): a closed form bounded by the area on every tangent plane. The catenoid between two rings is not a graph over a planar domain, and Theorem 3 does not apply to it; the computation below shows that for rings far enough apart the catenoid does not minimise area.

Theorem— Theorem 4 (the Plateau problem; Douglas 1931, Radó 1930)

Every Jordan curve $\Gamma \subset \R^3$ bounds a disc-type minimal surface: there is a map $f : \overline{D} \to \R^3$ , continuous, harmonic and conformal on the open disc $D$ , mapping $\partial D$ monotonically onto $\Gamma$ , whose image has least area among all disc-type surfaces spanning $\Gamma$ (assuming some spanning disc has finite area).

Proof sketch. Area is invariant under reparametrisation, so minimising it directly has no compactness. Douglas and Radó instead minimise the Dirichlet energy $E(f) = \tfrac12\int_D |\nabla f|^2 \ge \mathrm{Area}(f)$ , with equality exactly for conformal maps, over maps whose boundary values sweep out $\Gamma$ , normalised so that three fixed points of $\partial D$ go to three fixed points of $\Gamma$ . The three-point condition restores compactness (the Courant–Lebesgue lemma), a minimiser is harmonic, and minimising also over reparametrisations of the disc makes it conformal. The solution may a priori have branch points; Osserman (1970), Alt (1973) and Gulliver (1973) showed an area-minimising solution in $\R^3$ has none in the interior. See Colding–Minicozzi, Chapter 4, or Dierkes–Hildebrandt–Sauvigny.

Theorem— Theorem 5 (Bernstein's theorem and its limits)

An entire solution $u : \R^2 \to \R$ of the minimal surface equation is affine (Bernstein, 1915–17). The same holds for entire minimal graphs over $\R^n$ with $n \le 7$ (Fleming, De Giorgi, Almgren, Simons, 1962–68) and fails for $n \ge 8$ (Bombieri, De Giorgi and Giusti, 1969, using the Simons cone).

Theorem— Theorem 6 (harmonic maps; Eells–Sampson 1964)

If $M$ and $N$ are compact Riemannian manifolds and $N$ has non-positive sectional curvature, then every continuous map $M \to N$ is homotopic to a harmonic map, obtained as the limit of the harmonic map heat flow $\partial_t f = \tau(f)$ .

Without the curvature hypothesis the flow can blow up: for maps from surfaces, Sacks and Uhlenbeck (1981) showed that energy concentrates at points and bubbles off harmonic 2-spheres, and used this to produce minimal 2-spheres in every compact manifold with $\pi_2 \ne 0$ . And harmonic representatives need not exist: there is no harmonic map of degree $\pm1$ from the torus to the round 2-sphere (Eells–Wood, 1976).

Theorem— Theorem 7 (the Yamabe problem; Yamabe 1960, Trudinger 1968, Aubin 1976, Schoen 1984)

Every compact Riemannian manifold of dimension $n \ge 3$ admits a conformal metric of constant scalar curvature.

Writing the conformal metric as $\tilde g = \phi^{4/(n-2)} g$ , the problem becomes the semilinear equation $-\frac{4(n-1)}{n-2}\Delta\phi + R_g\phi = \lambda\phi^{(n+2)/(n-2)}$ , whose exponent is critical for the Sobolev embedding, so compactness fails. Yamabe's proof had a gap; Trudinger repaired it when the Yamabe invariant of the conformal class is non-positive, Aubin proved the case $n \ge 6$ (non-locally-conformally-flat), and Schoen completed the proof using the positive mass theorem. See Lee–Parker.

Theorem— Theorem 8 (Ricci flow; Hamilton 1982)

On a closed manifold, the Ricci flow has a unique smooth solution for a short time from any initial metric (Hamilton 1982; a simpler proof by DeTurck, 1983).
If a closed 3-manifold admits a metric of positive Ricci curvature, the normalised Ricci flow from that metric converges to a metric of constant positive sectional curvature; hence the manifold is diffeomorphic to a spherical space form $S^3/\Gamma$ .

The engine is the maximum principle. Under Ricci flow the scalar curvature evolves by $\partial_t R = \Delta R + 2|\mathrm{Ric}|^2$ .

Theorem— Proposition 9 (scalar maximum principle)

Along a Ricci flow on a closed manifold, $R_{\min}(t) = \min_M R(\cdot, t)$ is non-decreasing. In particular positive scalar curvature is preserved.

Proof

Given $\varepsilon > 0$ , let $\tilde R = R + \varepsilon(1 + t)$ ; then $\partial_t\tilde R = \Delta \tilde R + 2|\mathrm{Ric}|^2 + \varepsilon > \Delta\tilde R$ . Suppose $\tilde R$ first reaches a value below $m = \min_M \tilde R(\cdot, 0)$ — more precisely, let $t_0$ be the first time with $\min \tilde R(\cdot, t_0) = m - \delta$ for some small $\delta > 0$ , attained at $x_0$ . At $(x_0, t_0)$ we have $\Delta\tilde R \ge 0$ (spatial minimum) and $\partial_t\tilde R \le 0$ (the minimum has just decreased to this value), contradicting $\partial_t\tilde R > \Delta\tilde R$ . So $\tilde R \ge m - \delta$ for every $\delta$ , hence $R(\cdot, t) + \varepsilon(1 + t) \ge R_{\min}(0) + \varepsilon$ ; let $\varepsilon \to 0$ .

∎

The discrete analogue — an explicit finite-difference heat scheme never exceeds its initial bounds when the time step is small enough — is formalised below. Hamilton extended the principle to tensors (curvature pinching), which is how Theorem 8(2) is proved.

Theorem— Theorem 10 (Perelman 2002–2003; Ricci flow with surgery)

For every closed orientable 3-manifold there is a Ricci flow with surgery, defined for all time: singularities form only in regions modelled on shrinking necks $S^2 \times \R$ , caps, or round components, and are removed by cutting along 2-spheres and gluing in caps. The long-time behaviour yields the geometric decomposition; hence Thurston's geometrisation conjecture and the Poincaré conjecture hold.

The key new tools were Perelman's entropy functional, monotone along the flow, which gives a no local collapsing theorem, and his classification of ancient solutions (κ-solutions) that model singularities. Complete accounts: Morgan–Tian (2007), Kleiner–Lott (2008), Cao–Zhu (2006). For simply connected manifolds, finite extinction (Perelman; Colding–Minicozzi) gives a shorter route to the Poincaré conjecture. See low-dimensional topology.

Theorem— Theorem 11 (mean curvature and curve-shortening flow)

(Huisken 1984.) A closed convex hypersurface in $\R^{n+1}$ , $n \ge 2$ , shrinks under mean curvature flow to a point in finite time, becoming asymptotically round.
(Gage–Hamilton 1986, Grayson 1987.) A smooth embedded closed curve in the plane stays embedded, becomes convex, and shrinks to a point, asymptotically round, at time $A(0)/2\pi$ .

In higher dimensions embeddedness does not prevent singularities before extinction: Grayson's dumbbell (1989) pinches at its neck. The experiment below follows Theorem 11(2) numerically.

ExplanationH

The landscape: what is known and what is open

Established. Beyond the theorems above: the positive mass theorem (Schoen–Yau 1979 by minimal surfaces; Witten 1981 by spinors); Yau's solution of the Calabi conjecture (1978) via the complex Monge–Ampère equation; the Willmore conjecture, proved by Marques and Neves (2014) with min-max theory for minimal surfaces; Yau's conjecture that every closed 3-manifold contains infinitely many minimal surfaces (Irie–Marques–Neves 2018 for generic metrics, Song 2018 for all metrics); Lawson's conjecture that the Clifford torus is the only embedded minimal torus in the round 3-sphere (Brendle 2013); the differentiable sphere theorem (Brendle–Schoen 2009) via Ricci flow; and the existence of Kähler–Einstein metrics on K-stable Fano manifolds (Chen–Donaldson–Sun 2015).

Open problems.

Ricci flow in dimension 4 and higher: there is no analogue of Perelman's classification of singularities, and no general theory of surgery.
Singularities of mean curvature flow: for surfaces in $\R^3$ the generic singularities are spheres and cylinders (Colding–Minicozzi, 2012), and flows through neck-pinches are now well understood (Bamler–Kleiner and others), but a complete theory in higher dimensions is open.
Chern's conjecture (around 1970): a closed minimal hypersurface in the round sphere $S^{n+1}$ with constant scalar curvature has scalar curvature in a discrete set; open in general.
Yau's first eigenvalue conjecture (1982): for a closed embedded minimal hypersurface in $S^{n+1}$ , the first non-zero eigenvalue of the Laplacian equals $n$ ; known in special cases.

Formal mathematicsF

What Lean can say: catenoids, Scherk's surface, spheres and maximum principles

Mathlib has calculus, harmonic functions and the mean value property, but no mean curvature, minimal surfaces, harmonic maps between manifolds, Ricci curvature or Ricci flow. So we formalise the computations that the theorems rest on, each stated as a precise analytic fact. The catenoid (Proposition 2):

catenary_euler_lagrangeLean 4Proved here from basic factsFormally verified

/-- **The catenoid is a minimal surface of revolution.** Rotating the graph of `y(x) > 0` about
the `x`-axis gives area `2π ∫ y √(1 + y'²) dx`, whose Euler–Lagrange equation is
`y y'' = 1 + y'²`. The catenary `y = a cosh(x/a)` solves it: `y' = sinh(x/a)`,
`y'' = cosh(x/a)/a`, and `y y'' = cosh²(x/a) = 1 + sinh²(x/a)`. -/
theorem catenary_euler_lagrange (a : ℝ) (ha : a ≠ 0) (x : ℝ) :
    HasDerivAt (fun x => a * cosh (x / a)) (sinh (x / a)) x ∧
      HasDerivAt (fun x => sinh (x / a)) (cosh (x / a) / a) x ∧
      a * cosh (x / a) * (cosh (x / a) / a) = 1 + sinh (x / a) ^ 2 := by
  have hdiv : HasDerivAt (fun x : ℝ => x / a) (1 / a) x := by
    simpa using (hasDerivAt_id x).div_const a
  refine ⟨?_, ?_, ?_⟩
  · have h := ((hasDerivAt_cosh (x / a)).comp x hdiv).const_mul a
    exact h.congr_deriv (by field_simp)
  · have h := (hasDerivAt_sinh (x / a)).comp x hdiv
    exact h.congr_deriv (by field_simp)
  · field_simp
    linarith [cosh_sq (x / a)]