Theory of metric Lie groups andH-surfaces in homogeneous3-manifolds. William H. Meeks III University of Massachusetts at Amherst Based on joint work with Mira, P´erez,Ros and Tinaglia. Definition A 2-dimensional submanifold with constant mean curvature H ≥ 0 in a Riemannian 3-manifold is called an H-surface. Definition If the isometry group of a Riemannian manifold Y acts transitively, then Y is called homogeneous. Definition A Lie group with left invariant metric is called a metric Lie group. Notation and Language Y = simply connected homogeneous 3-manifold. X = simply connected 3-dimensional Lie group with left invariant metric (X is a metric Lie group). H(Y) = Inffmax jHMj : M = immersed closed surface in Yg, where max jHMj denotes max of absolute mean curvature function HM. The number H(Y) is called the critical mean curvature of Y. Area(@K) Ch(Y) = Inf = Cheeger constant of Y: K⊂Y compact Volume(K) Goals of Lecture 1 Classify simply connected homogeneous 3-manifolds Y. Present the basic theory of simply connected, 3-dimensional metric Lie groups X. Express metric and bases for left and right invariant vector fields in 2 explicit coordinates for X = R oA R = a semidirect product. Give formulae for the Levi-Civita connection r, the principal Ricci curvature vectors and the principal Ricci curvature values for X. Goals of Lecture 2 Classification and quasi-isometric classification of the possible Y. Right cosets of 2-dimensional subgroups H ⊂ X and the existence of algebraic open book decompositions. Uniqueness and embeddedness of minimal spheres in X ≈ SU(2). Isoperimetric domains and the isoperimetric profile of Y. Explain the formula: Ch(Y) = 2H(Y) for non-compact Y. Discuss CMC foliations, Isoperimetric Inequality Conjectures, Stability Conjecture, Product CMC Foliation Conjectures. Goals of Lecture 3 Classification of algebraic open book decompositions. The left invariant Lie group Gauss map and the embeddedness problem for H-spheres. The Meeks-Tinaglia curvature and radius estimates for H-disks. Existence and uniqueness of H-spheres in X (Meeks-Mira-P´erez-Ros). Discuss various conjectures on embedded H-surfaces in Y. Goals of Lecture 4 The H-potential for X and a Weierstrass representation for H-surfaces in X (Meeks-Mira-P´erez-Ros). Outline of the proof of the existence and uniqueness of H-spheres in X (Meeks-Mira-P´erez-Ros). The Plateau problem in X. Other theoretical techniques and open problems. Theorem (Simply connected homogeneous 3-dimensional Y) If Y is a simply connected homogenous 3-manifold, then: Y is isometric to a metric Lie group (Lie group with left invariant metric); these examples form a 3-parameter family of non-isometric homogeneous 3-manifolds. or 2 Y is isometric to S (κ) × R for some κ > 0. Theorem (Simply connected 3-dimensional Lie groups X) Every X is isomorphic to SU(2) = fgroup of unit length quaterionsg or to the universal covering group of a 3-dimensional subgroup of the 6-dim affine group 2 2 2 F = ff(x) = Ax + b: R ! R j b 2 R ; A 2 GL(2; R)g, which is the natural 2 2 semidirect product of R with GL(2; R) = Aut(R ). The 3-dimensional subgroups of F are one of the following types: SL(2; R) = fA 2 GL(2; R) j det(A) = 1g. 2 The semidirect product of the normal subgroup R ⊂ F of translations with any particular 1-parameter subgroup Γ of GL(2; R). Remark These two theorems will be proved at the beginning of Lecture 2. Definition A Lie groupG is a smooth manifold whose group operation ∗ satisfies the following: ∗: G × G ! G (a; b) 7−! a ∗ b I : G ! G a 7! a−1 are both smooth mappings. We will frequently use the multiplicative notation ab to denote a ∗ b. The respective left and right multiplications by a 2 G are given by: la : G ! G x 7−! ax ra : G ! G x 7−! xa: Given a; p 2 G, let (la)∗ : TpG ! TapG (resp (ra)∗) denote the differential of la (resp ra). For vp 2 TpG, let avp (resp. vpa) denote the vector (la)∗(vp) 2 TapG (resp. vpa = (ra)∗(vp) 2 TpaG). Definition A vector field X is called left invariant if 8a 2 G, X = aX , or equivalently, for each p 2 G, Xap = aXp: A vector field Y is called right invariant if 8a 2 G, Y = Ya. The vector space L(G) of left invariant vector fields on G can be identified with the tangent space Te G at the identity element e 2 G. L(G) is a Lie algebra under the Lie bracket of vector fields, i.e., for X ; Y 2 L(G), then [X ; Y ] 2 L(G): L(G) is called the Lie algebra of G. Remark (1-parameter subgroups) For each X 2 L(G), the integral curve G(X ) passing through the identity is the image of a 1-parameter subgroup, i.e., there is a Lie group homomorphism expXe : R ! G(X ) ⊂ G . In this case X is the velocity vector exp0 (0) of exp (t) at e. e Xe Xe 2 Let M(n; R) = Rn be the set of real n × n matrices, which is a Lie algebra under the operation [A; B] = AB − BA. 2 When G is a subgroup of GL(n; R) ⊂ M(n; R) = Rn , then L(G) = Te G can be identified with the subspace of M(n; R) = Te GL(2; R) of tangent vectors to G at the identity matrix e = In. When G ⊂ GL(2; R) and Xe = A 2 M(n; R) = Te G; 1 X tnAn exp (t) = exp(tA) = 2 G(X ) ⊂ G: Xe n! n=0 This explains the notation for the group homomorphism expXe : R ! G, for X 2 L(G). Theorem (Ado's Theorem) Given a n-dimensional real Lie algebra L, then for some k 2 N, L is isomorphic to a subalgebra L0 of the Lie algebra of k × k matrices M(k; R) under the bracket of matrices. L is isomorphic to the Lie algebra of the connected Lie subgroup G of GL(k; R) generated by the elements 1 X tnAn exp(tA) = 2 GL(k; ) for t 2 ; A 2 L0: n! R R n=0 Therefore, every simply connected Lie group is the universal covering group of some subgroup G ⊂ GL(k; R) for some k 2 N. Definition Given an X 2 L(G) with related 1-parameter subgroup G(X ) ⊂ G, then X is the velocity vector field associated to the 1-parameter group of diffeomorphisms obtained by letting G(X ) act on G from the right. The velocity field of the 1-parameter group of diffeomorphisms obtained on G by the left action of G(X ) on G is the right invariant X X vector field K , where Ke = Xe . Let R(G) denote the n-dimensional vector space of right invariant vector fields, where n is the dimension of G. R(G) is also a Lie algebra under the Lie bracket of vector fields. The vector field K X is the Killing field associated to the 1-parameter group of isometries G(X ) for ANY left invariant metric on G. K(X) = space of Killing fields in X is a Lie algebra isomorphic to the Lie algebra of the isometry group Iso(X) of X. Definition (And Facts) A Riemannian metric on Lie group G is called left invariant if for all a 2 G; la : G ! G is an isometry of G. Each left invariant metric on G is obtained by taking a metric h; ie on Te G and defining for a 2 G and v; w 2 TaG; −1 −1 hv; wi = ha v; a wie : We call a Lie group Y with a left invariant metric, a metric Lie group. The related metric Lie group is complete. The universal cover Ye with the pulled back metric is a metric Lie group and Y = Ye/π1(Y), where we identify π1(Y) with a discrete normal abelian subgroup of Y acting on Y by left translation. Example (The metric Lie group R) The real numbers R with its usual metric and group operation + is a Lie group with a left invariant metric. In this case L(R) = R(R) are just the constant vector fields t Xp = (p; t), where we consider the tangent bundle TR of R to be R × R. 1 By taking X = (0; 1) 2 T0 , exp 1 : ! is a group 0 R X0 R R isomorphism, which is essentially the identity map. Theorem (Simply connected homogeneous 1 and 2-manifolds) R with its usual metric is the unique simply connected homogeneous 1-manifold. Every simply connected homogeneous 2-manifold has constant sectional curvature τ 2 R and is isometric to R2 for τ = 0, to S2(τ) for τ > 0, or to the "hyperbolic plane" H(τ) for τ < 0. Except for S2(τ), the simply connected homogeneous manifolds of dimension ≤ 2 are isometric to metric to Lie groups. Example (The metric Lie group H = R oA=(1) R) Here we consider A = (1) to be the 1 × 1 identity matrix. The notation R oA=(1) R means we have a semidirect product of R with R via the homomorphism σ : R ! Aut(R) = R − f0g; where At t σ(t) = e = e . Let σt denote the automorphism σ(t). Then the non-commutative group operation ? on H is b (a; b) ? (c; d) = (a + σb(c); b + d) = (a + e c; b + d): Choose the metric at (Te=(0;0)H) = R × R to be the usual one, i.e., j@x j(0; 0) = j@t j(0; 0) = 1 and extend it to a left invariant metric.