Results on isometric immersions of surfaces, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Curvature, Hessians and immersions under affine changes of coordinates — Lemma 1.2. For the pull-back g' = M^T (g o Phi) M of a metric by an affine map Phi(x) = c + Mx: the curvature is K o Phi, the covariant Hessian of z o Phi is M^T (H o Phi) M, |d(z o Phi)| = |dz| o Phi, and F o Phi is an isometric immersion if F is.
• The height equation of an isometric immersion — Theorem 2.1. For an isometric immersion F of a plane domain with metric g into R^3 and a unit vector e, the height z = <F, e> satisfies Hess_g z = <n, e> II, |dz|^2 + <n, e>^2 = 1 and det Hess_g z = K det g (1 - |dz|^2).
• Curvature of dX^2 + f^2 dY^2 — Lemma 3.1. The metric dX^2 + f(X, Y)^2 dY^2 with f > 0 has the curvature -f_XX / f.
• A metric with a given small curvature — Proposition 3.2. For every smooth function K_0 on the square (-1, 1)^2 with |K_0| <= 1/1000 there is a smooth f with |f - 1| < 1/1000 such that dX^2 + f^2 dY^2 has the curvature K_0.
• A boundary point with a second fundamental form of rank one — Proposition 4.2. If the curvature is negative in a round disk and zero on its boundary, and an isometric immersion of a neighbourhood is a graph over a plane, then at some point of the boundary circle the second fundamental form has rank one.
• A symmetric form of rank one in a rotated frame — Lemma 2.1. For a symmetric bilinear form B of rank one on the plane and an orthonormal basis r_1, r_2 whose first line makes the angle alpha with the kernel, |alpha| < pi/2: B(r_2, r_2) is not zero and |B(r_1, r_2) / B(r_2, r_2)| = |tan alpha|.
• Placement of the patches — Lemma 3.1. There are pairwise disjoint squares c_n + lambda_n R_n [-4, 4]^2, with rotations from a finite set, outside the closed disks D_j and inside the doubled disks, such that for every boundary point of every D_j and every rotation of the set, squares with that rotation lie in arbitrarily small disks around the point.
• The assembled metric (assuming the patch obstruction) — Proposition 4.3. Assuming (P): the sum of the background metric and the pushed-forward perturbations is a smooth metric on (-1, 1)^2; its pull-back to each patch is a model metric plus a perturbation without admissible height; its curvature is negative in every disk D_j and zero on the boundary circles.
• A smooth metric with no local isometric immersion (assuming the patch obstruction) — Theorem 5.1. Assuming (P): there is a smooth metric on (-1, 1)^2 such that no open neighbourhood of the origin has a smooth isometric immersion into R^3.
• The Taylor series of the metric at the origin (assuming the patch obstruction) — Corollary 6.1. Assuming (P): the metric of Theorem 5.1 has the same derivatives of all orders at the origin as the Euclidean metric.
• The curvature tensor of a metric on a plane domain — Lemma 2.1. In dimension two, R_kji^m = K (delta^m_k g_ji - delta^m_j g_ki), where R is formed from the Christoffel symbols and K is the curvature.
• Commutation of third covariant derivatives of a function — Lemma 2.2. For a function z on a plane domain with a metric, the covariant derivative T of the covariant Hessian satisfies T_ijk - T_ikj = K (g_ik z_j - g_ij z_k).
• The drift identity for a solution of the height equation — Theorem 3.1. For a solution z of the height equation with H_yy not zero, q = H_xy/H_yy, D = d_x - q d_y and G = E/H_yy^2: Dq - q Gamma^x_DD - Gamma^y_DD = K_y G + K rho_1 with an explicit rho_1, and Dq - P_{z_x} q - P_{z_y} = K_y G + K (rho_1 - rho_2).
• The two leading terms of a high derivative of a composite function — Lemma 4.1. For l >= 3, the l-th derivative of F(y, p(y)) is sum F_{p_i} p_i^(l) + l sum (d/dy F_{p_i}) p_i^(l-1) plus terms in which all derivatives of p have order at most l - 2 and total order at most l.
• The equation for a high derivative of a solution of the height equation — Theorem 5.1. For l >= 3, u = d_y^l z satisfies D^2 u + a u_yy + B Du + ((l + 1) K_y G + K rho) u_y = R_l with a = K G, B = -P_{z_x} - 2 l q_y and a remainder R_l in which all derivatives of z have order at most l.
• The differentiated height equation in flow coordinates — Theorem 6.2. In coordinates t, s in which D = d_t, u = d_y^l z satisfies u_tt + A u_ss + B u_t + C u_s = R_l with A = G_1 K, G_1 = G sigma^2 > 0, and C = (l + 1) A_s + K (sigma rho + G sigma sigma_s - (l + 1) (G_1)_s).
• The multiplier identity for u_tt + A u_ss + B u_t + C u_s = f — Lemma 2.1. For multipliers m, n vanishing on the boundary of a rectangle, the integral of f (m u_s + n u_t) equals the integral of Q_tt u_t^2 + Q_ss u_s^2 + Q_ts u_t u_s with Q_tt = (m_s - n_t)/2 + B n, Q_ss = -(A m)_s/2 + (A n)_t/2 + C m, Q_ts = -m_t + B m - (A n)_s + C n.
• Positivity of the weighted quadratic form where A is negative or small — Proposition 3.1. Under (A1) and (A3), with the weight W = d^8 e^(-lambda s + I), m = -W, n = epsilon t W and explicit lambda and c_1, the quadratic form of the multiplier identity is at least c_2 W (xi^2 + eta^2) at the points with A <= c_1 d.
• A weighted elliptic estimate up to the free edge — Proposition 4.1. Under (A1), the integral of d^7 |grad u|^2 over the set where A >= c_1 d and the cut-off is 1 is bounded by a constant times the integral of u^2 + f^2.
• The estimate at the free edge for an equation of mixed type — Theorem 5.1. Let u_tt + A u_ss + B u_t + C u_s = f on a rectangle, with C = (l + 1) A_s + A omega and bounded coefficients (A1), A >= alpha_0 near three edges (A2), and -(l + 1/2) A_s + (epsilon/2) t A_t >= theta_0 where |A| <= a_0 (A3). Then on every smaller rectangle that keeps a distance from the fourth edge, the integral of u_t^2 + u_s^2 is bounded by a constant times the integral of u^2 + f^2 over the rectangle; the constant depends only on the numbers in the assumptions.
• A Sobolev inequality on a square — Lemma 1.1. For a smooth function w on the square p + [-r, r]^2: |w(p)|^2 <= r^-2 int w^2 + 2 int (w_x^2 + w_y^2) + 4 r^2 int w_xy^2.
• Flow coordinates of an admissible height on the square [-2, 2]^2 — Lemma 2.1. For an admissible height with bounds of order eight, the flow of D = d_x - q d_y defines coordinates t, s on [-2, 2]^2 with |y - s| <= 1/50, a bounded Jacobian factor sigma and bi-Lipschitz bounds, all depending only on the bounds.
• The differentiated height equation on a cap satisfies the assumptions of the estimate at the free edge — Proposition 3.1. On rectangles [-T, T] x [s_0, b] in the lower cap of the model square, the coefficients of the equation for u = d_y^l z satisfy (A1), (A2), (A3) with numbers depending only on the profile h, kappa, the bounds of order eight and l.
• The remainder of the differentiated height equation is bounded in mean square — Lemma 4.2. For l >= 8, if the derivatives of z of order at most l are bounded in mean square on a larger rectangle of the cap and the metric has bounds of every order there, then the remainder R_l, and the remainders in the formulas that express the derivatives with more x-indices, are bounded in mean square.
• Cap regularity for admissible heights — Theorem 5.1. For a family of admissible heights on the model square with common bounds of order eight, common bounds for all derivatives of the metrics on the images of the rectangles of the lower cap give common bounds for all derivatives of the heights there. The same holds for the upper cap (Corollary 5.2).
• The height equation as a hyperbolic equation for derivatives along xi — Proposition 1.2. In sheared coordinates xi = x, theta = y + q_0 x, on a set where K < 0, -a >= v_*^2 and |q - q_0| <= 1/(10 L): v = d_xi^j z satisfies v_theta theta - P_1 v_theta xi - S_1 v_xi xi = b^theta v_theta + b^xi v_xi + R with P_1 = 2(q - q_0)/d_0, S_1 = -1/d_0 > 0, |P_1| + sqrt(S_1) <= L/4, and a remainder built from derivatives along xi of z_xi and z_theta of order at most j - 1.
• The energy inequality on a shrinking interval — Lemma 2.1. For v_theta theta - P_1 v_theta xi - S_1 v_xi xi = b^theta v_theta + b^xi v_xi + R on a trapezoid whose sides move inwards with a speed Lambda >= |P_1| + sqrt(S_1), with S_1 >= s_min > 0 and bounded coefficients: sqrt(E(theta)) <= e^(c_E (theta - theta_0)) (sqrt(E(theta_0)) + integral of the L^2 norm of R), c_E = 3 beta_1 (1 + 1/s_min).
• A Sobolev inequality on an interval — Lemma 3.1. For a smooth function phi on an interval J and x_0 in J: phi(x_0)^2 <= (2/|J|) int phi^2 + 2 |J| int phi'^2.
• Propagation of bounds for all derivatives where the curvature is negative — Proposition 4.1. For a family of admissible heights with common low bounds, on a trapezoid in a hyperbolic region where the metrics have common bounds of every order: common mean-square bounds for all derivatives along xi of z_xi and z_theta on the initial line give common bounds for these derivatives on the whole trapezoid.
• The theorem of Baire for an open subset of a complete metric space — Theorem 1.2. If countably many subsets of a nonempty open subset U of a complete metric space are nowhere dense in U, every nonempty open subset of U contains a point that lies in none of them.
• The patch obstruction — Theorem 1.3. Let g_0 be a metric on a neighbourhood of [-3, 3]^2 with curvature kappa (x^2 - h(y)). Among the smooth perturbations eta supported in [-1/10, 1/10]^2 for which g_0 + eta is positive definite with curvature below -kappa/2 on [-1/5, 1/5]^2, those for which g_0 + eta has no admissible height are dense for the convergence of all derivatives.
• Bounds for an admissible height at the edges of the oscillating strip — Proposition 3.1. For the heights of a class with fixed bounds and metrics that oscillate only in the strip |theta| <= delta/tau: all derivatives along xi of z and z_theta are bounded at theta = +-delta/tau, independently of tau.
• The second-order part of K det g — Lemma 5.1. K det g = d_1 d_2 g_12 - (1/2) d_1^2 g_22 - (1/2) d_2^2 g_11 plus a polynomial in the first derivatives of g with coefficients rational in g.
• The forcing by the oscillating perturbation — Lemma 5.2. The difference of the solved height equations for the perturbed and the unperturbed metric is tau^(2-N) D_tau chi_0 phi_0 cos(tau xi) up to O(tau^(1-N)), with D_tau = (1 - |dz|^2) / (2 H_xi xi) of one sign and between two positive constants.

Reading order

Seven lessons with complete proofs. The second lesson proves the theorem from the patch obstruction, which the seventh lesson proves, using lessons three to six. The lessons can be read in order; the first two give the theorem and its geometric part, the other five the analysis.