By Tobias Holck Colding, William P. Minicozzi II

Minimum surfaces date again to Euler and Lagrange and the start of the calculus of adaptations. a number of the thoughts constructed have performed key roles in geometry and partial differential equations. Examples comprise monotonicity and tangent cone research originating within the regularity concept for minimum surfaces, estimates for nonlinear equations in response to the utmost precept bobbing up in Bernstein's classical paintings, or even Lebesgue's definition of the crucial that he built in his thesis at the Plateau challenge for minimum surfaces. This ebook begins with the classical thought of minimum surfaces and finally ends up with present study issues. Of a few of the methods of coming near near minimum surfaces (from complicated research, PDE, or geometric degree theory), the authors have selected to target the PDE features of the idea. The ebook additionally includes a number of the functions of minimum surfaces to different fields together with low dimensional topology, common relativity, and fabrics technology. the one necessities wanted for this booklet are a uncomplicated wisdom of Riemannian geometry and a few familiarity with the utmost precept

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In our case, for x ∈ X, ρt (x) = x (all t) is the constant curve, so vt vanishes at all x for all t, hence µx = 0. To check that Q above satisfies the homotopy formula, we compute 1 Qdω + dQω 1 ρ∗t (ıvt dω)dt + d = 0 ρ∗t (ıvt ω)dt 0 1 ρ∗t (ıvt dω + dıvt ω )dt , = 0 Lvt ω where Lv denotes the Lie derivative along v (reviewed in the next section), and we used the Cartan magic formula: Lv ω = ıv dω + dıv ω. The result now follows from d ∗ ρ ω = ρ∗t Lvt ω dt t and from the fundamental theorem of calculus: 1 Qdω + dQω = 0 d ∗ ρ ω dt = ρ∗1 ω − ρ∗0 ω .

With this form, the total area of S 2 is 4π. Consider cylindrical polar coordinates (θ, z) on S 2 away from its poles, where 0 ≤ θ < 2π and −1 ≤ z ≤ 1. Show that, in these coordinates, ω = dθ ∧ dz . 2. Prove the Darboux theorem in the 2-dimensional case, using the fact that every nonvanishing 1-form on a surface can be written locally as f dg for suitable functions f, g. Hint: ω = df ∧ dg is nondegenerate ⇐⇒ (f, g) is a local diffeomorphism. 3. Any oriented 2-dimensional manifold with an area form is a symplectic manifold.

Suppose that M is compact. , {exp tv : M → M | t ∈ R} is the unique smooth family of diffeomorphisms satisfying exp tv|t=0 = idM and d (exp tv)(p) = v(exp tv(p)) . 3 The Lie derivative is the operator Lv : Ωk (M ) −→ Ωk (M ) defined by d (exp tv)∗ ω|t=0 . dt Lv ω := When a vector field vt is time-dependent, its flow, that is, the corresponding isotopy ρ, still locally exists by Picard’s theorem. More precisely, in the neighborhood of any point p and for sufficiently small time t, there is a one-parameter family of local diffeomorphisms ρt satisfying dρt = vt ◦ ρt dt and ρ0 = id .

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