Hsiang algebras of cubic minimal cones Vladimir G. Tkachev Linköping University November 4, 2016 Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (1 of 26)
Dedicated to Sasha (1 April 1962 19 October 2016) Zürich, ICM94 Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (2 of 26)
Minimal cones and singular solutions Evans, Crandall, Lions, Jensen: Given uniformly elliptic operator F, the Dirichlet problem F (D 2 u)= 0 has a unique viscosity solution. Trudinger, Caffarelli, early 80 s: the solution is always C 1,ε Nirenberg, 50 s: if n = 2 then u is classical (C 2 ) solution Nadirashvili, Vlǎduţ, 2007: there are solutions which are not C 2 for n = 24. Theorem (Nadirashvili, Vlăduţ, V.T., 2012) The function w(x) := u 1(x) x where u 1 (x) = x 3 5 + 3 2 x 5(x 2 1 + x2 2 2x2 3 2x2 4 ) + 3 3 2 x 4(x 2 2 x2 1 ) + 3 3x 1 x 2 x 3, is a singular viscosity solution of the uniformly elliptic Hessian equation ( w) 5 + 2 8 3 2 ( w) 3 + 2 12 3 5 w + 2 15 det D 2 (w) = 0. Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (3 of 26)
Hsiang s Problems W.-Y. Hsiang (J. Diff. Geometry, 1, 1967): Let u be a homogeneous polynomial in R n. Then u 1 (0) is a minimal cone iff 1u := Du 2 u 1 2 Du, D Du 2 0 mod u. In deg = 2: {(x, y) R k+m : (m 1) x 2 = (k 1) y 2 } The first non-trivial case: deg u = 3 and then 1u = a quadratic form u(x) (1) In fact, all known irreducible cubic minimal cones satisfy very special equation: 1u = λ x 2 u(x) (2) Problem 1: Classify all cubic minimal cones, or at least all solutions of (2). Problem 2: Are there irreducible minimal cones in R m of arbitrary high degree? Problem 3: Prove that any closed minimal submanifold M n 1 S m is algebraic. Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (4 of 26)
Hsiang s Problems A homogeneous cubic form u(x) is called a Hsiang cubic if 1u = λ x 2 u(x), λ R. Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (5 of 26)
What about the four nontrivial solutions? Hsiang s trick: let X H k (A) = trace free hermitian k k-matrices over A = R or C 1 is an O(n)-invariant 1(tr X 3 ) = is a polynomial in tr X 2,..., tr X k deg( 1 tr X 3 ) = 5 if 3 k 4 then 1u(X) = c 1 tr X 2 tr X 3 = c 1 X 2 u(x). Thus u = tr X 3 is a Hsiang cubic. This yields the four Hsiang examples w in H 3 (R) = R 5, H 3 (C) = R 8, H 4 (R) = R 9, H 4 (C) = R 15 An important observation: deg w = 3 implies tr(d 2 u) = 0 the harmonicity of u tr(d 2 u) 2 = C 1 x 2 the quadratic trace identity tr(d 2 u) 3 = C 2u the cubic trace identity Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (6 of 26)
What about the four nontrivial solutions? Hsiang s trick: let X H k (A) = trace free hermitian k k-matrices over A = R or C 1 is an O(n)-invariant 1(tr X 3 ) = is a polynomial in tr X 2,..., tr X k deg( 1 tr X 3 ) = 5 if 3 k 4 then 1u(X) = c 1 tr X 2 tr X 3 = c 1 X 2 u(x). Thus u = tr X 3 is a Hsiang cubic. This yields the four Hsiang examples w in H 3 (R) = R 5, H 3 (C) = R 8, H 4 (R) = R 9, H 4 (C) = R 15 An important observation: deg w = 3 implies tr(d 2 u) = 0 tr(d 2 u) 2 = C 1 x 2 tr(d 2 u) 3 = C 2u the harmonicity of u the quadratic trace identity the cubic trace identity Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (7 of 26)
How do nonassociative algebras enter? u = Re(z 1z 2)z 3, z i A d, d = 1, 2, 4, 8, the triality polynomials in R 3d where A 1 = R, A 2 = C, A 4 = H, A 8 = O are the classical Hurwitz algebras. 1 3 x 1 + x 2 x 3 x 4 2 u(x) = x 2 3 x 1 x 5 = a Cartan isoparametric cubic in R 5 1 x 4 x 5 3 x 1 x 2 u(x) = x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 So, Hsiang minimal cubics can be interpreted very nicely certain algebraic structures. Which ones and how? Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (8 of 26)
Hsiang cubics of Clifford type Example. The Lawson cubic cone in R 4 with the defining polynomial u(z) = (x 2 1 x 2 2)y 1 + 2x 1x 2y 2 = x, A 1x y 1 + x, A 2x y 1, z = (x, y) R 4 A 1 = ( 1 0 0 1 ) ( 0 1, A 2 = 1 0 ) Theorem (V.T., 2010) Let {A i} 1 1 q be a symmetric Clifford system, i.e. A 2 i = I and A ia j + A ja i = 0, i j. Then u A(z) = q i=1 z = (x, y) R2p R q is a Hsiang cubic. The existence of a symmetric Clifford system in R 2p is equivalent to q 1 ρ(p), ρ(p) = Hurwitz-Radon function = 1 + #(of independent vector fields on S p 1 ) Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (9 of 26)
The dichotomy of Hisang cubics A Hsiang cubic u is of Clifford type if u = u A, otherwise it is called exceptional. Theorem (V.T., 2010) Hsiang cubics of Clifford type are congruent if and only if the corresponding symmetric Clifford systems are geometrically equivalent. representation theory of Clifford algebras yields a complete classification of Hsiang cubics of Clifford type. Main Problem: How to determine all exceptional Hsiang cubics? Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (10 of 26)
Exceptional Hsiang cubics The Hsiang examples u in R 5, R 8, R 9, R 15 are exceptional Hsiang cubics. Proof. By contradiction: is u is the Clifford type then u = u A = q i=1 tr(d2 u A) 2 = 2q x 2 + 2p y 2. But the quadratic trace identity yields tr(d 2 u) 2 = C 1( x 2 + y 2 ), implying by the O(n)-invariance of 1 that q = p. Since q 1 ρ(p) p {1, 2, 4, 8}, thus n = q + 2p {3, 6, 12, 24}, a contradiction. Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (11 of 26)
The nonassociative algebra approach Given a cubic form u on an inner product vector space (V,, ) define a composition law (x, y) xy as the unique element satisfying u(x, y, z) = xy, z, z V. This defined algebra is commutative, nonassociative and metrized and in this setting, u(x) = 1 6 x, x2 Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (12 of 26)
The nonassociative algebra approach Given a cubic form u on an inner product vector space (V,, ) define a composition law (x, y) xy as the unique element satisfying u(x, y, z) = xy, z, z V. This defined algebra is commutative, nonassociative and metrized and in this setting, u(x) = 1 6 x, x2 Du(x) = 1 2 x2 Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (12 of 26)
The nonassociative algebra approach Given a cubic form u on an inner product vector space (V,, ) define a composition law (x, y) xy as the unique element satisfying u(x, y, z) = xy, z, z V. This defined algebra is commutative, nonassociative and metrized and in this setting, u(x) = 1 6 x, x2 Du(x) = 1 2 x2 xy = (D 2 u(x))y Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (12 of 26)
The nonassociative algebra approach Given a cubic form u on an inner product vector space (V,, ) define a composition law (x, y) xy as the unique element satisfying u(x, y, z) = xy, z, z V. This defined algebra is commutative, nonassociative and metrized and in this setting, u(x) = 1 6 x, x2 Du(x) = 1 2 x2 xy = (D 2 u(x))y L x = D 2 u(x), i.e. the multiplication operator by x is the Hessian of u at x Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (12 of 26)
The nonassociative algebra approach Given a cubic form u on an inner product vector space (V,, ) define a composition law (x, y) xy as the unique element satisfying u(x, y, z) = xy, z, z V. This defined algebra is commutative, nonassociative and metrized and in this setting, u(x) = 1 6 x, x2 Du(x) = 1 2 x2 xy = (D 2 u(x))y L x = D 2 u(x), i.e. the multiplication operator by x is the Hessian of u at x L x is self-adjoint: L xy, z = y, L xz (cf. the Freudenthal-Tits-Springer construction of exceptional Jordan algebras) Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (12 of 26)
The nonassociative algebra approach a cubic form u satisfying a PDE a metrized algebra V (u) with an identity Examples: The Laplace equation: The eiconal (Cartan-Münzner) equation: u(x) = 0 = tr L x = 0 Du(x) 2 = 9 x 4 = x 2 2 = 36 x 4 The third fundamental form XA(Y, Z) of an isoparametric hypersurface in a space form Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (13 of 26)
Key steps of the proof The set of idempotents Ide(V ) is nonempty: any stationary point of u(x) on S n 1 gives rise to an idempotent. Given c Ide(V ), L c is a self-adjoint the Peirce decomposition V = k V c(t α), V c(t α) := ker(l c t α) α=1 Use the defining PDE to capture the multiplication table : V c(t α)v c(t β ) γ V c(t γ) If the PDE is good enough, there are some natural (Clifford or Jordan) algebra structures hidden inside V. The tetrad decomposition Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (14 of 26)
The nonassociative algebra approach Now, let u(x) be a Hsiang cubic, i.e. Du(x) 2 u(x) 1 2 Du(x), D Du(x) 2 = λ x 2 u(x). Then the corresponding Freudenthal-Springer algebra satisfies x 2, x 2 tr L x x 2, x 3 = 2 3 λ x, x x2, x Definition. A commutative metrized algebra is called Hsiang if the latter identity satisfied. Hsiang cubics Hsiang algebras Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (15 of 26)
Key steps of the proof A Hsiang algebra := trivial if dim V V = 1 (or u = a, x 3 ). Theorem A (The Harmonicity) Any nontrivial Hsiang algebra is harmonic: tr L x = 0 and λ 0. In particular, x 2, x 3 = 4 3 x2, x x 2. Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (16 of 26)
Key steps of the proof A Hsiang algebra := trivial if dim V V = 1 (or u = a, x 3 ). Theorem A (The Harmonicity) Any nontrivial Hsiang algebra is harmonic: tr L x = 0 and λ 0. In particular, x 2, x 3 = 4 3 x2, x x 2. What about Hsiang cubics of Clifford type? Theorem B u is a Hsiang cubic of Clifford type iff V (u) admits a non-trivial Z 2-grading V = V 0 V 1, V 0V 0 = 0 and x V 0: L 2 x = x 2 on V 1. Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (16 of 26)
Key steps of the proof Theorem C (The hidden Clifford algebra structure) Let V be a Hisang algebra. Then (i) c Ide(V ), the associated Peirce decomposition is V = V c(1) V c( 1) V c( 1 ) Vc( 1 ) and dim Vc(1) = 1; 2 2 (ii) The Peirce dimensions n 1 = dim V c( 1), n 2 = dim V c( 1 2 ) and n3 = dim Vc( 1 2 ) do not depend on a particular choice of c and n 3 = 2n 1 + n 2 2; (iii) If ρ is the Hurwitz-Radon function then n 1 1 ρ(n 1 + n 2 1). In particular, for each n 2 there exist only finitely many possible values of n 1. Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (17 of 26)
Key steps of the proof Theorem D (The Multiplication Table) If V 0 = V c(1), V 1 = V c( 1), V 2 = V c( 1 2 ), V3 = Vc( 1 2 ) then V 0 V 1 V 2 V 3 V 0 V 0 V 1 V 2 V 3 V 1 V 1 V 0 V 3 V 2 V 3 V 2 V 2 V 3 V 0 V 2 V 1 V 2 V 3 V 3 V 2 V 3 V 1 V 2 V 0 V 1 V 2 In particular, V 0 V 1 and V 0 V 2 are subalgebras of V. Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (18 of 26)
Jordan algebras An algebra V with a commutative product is called Jordan if [L x, L x 2] = 0 x V. Main examples 1) The Jordan algebra H n(a d ) of Hermitian matrices of order n, d = 1, 2, 4 with x y = 1 (xy + yx) 2 2) The spin factor S (R n+1 ) with (x 0, x) (y 0, y) = (x 0y 0 + x, y ; x 0y + y 0x) Theorem (Jordan-von Neumann-Wigner, 1934) Any finite-dimensional formally real Jordan algebra is a direct sum of the simple ones: the spin factors S (R n+1 ); the Jordan algebras H n(a d ), n 3, d = 1, 2, 4; the Albert algebra H 3(A 8). Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (19 of 26)
Key steps of the proof Theorem E (The hidden Jordan algebra structure) Let V be a Hisang algebra and c Ide(V ). Then the subspace J c := V c(1) V c( 1 2 ) carries a structure of a formally real rank 3 Jordan algebra, and the following conditions are equivalent: (i) the Hsiang algebra V is exceptional; (ii) J c is a simple Jordan algebra; (iii) n 2 2 and the quadratic trace identity tr L 2 x = k x 2 holds for some k R. The proof of the first part of the theorem is heavily based on the McCrimmon-Springer construction of a cubic Jordan algebra. Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (20 of 26)
The Finiteness of Exceptional Hsiang Algebras If V is an exceptional Hsiang algebra then J c = V c(1) V c( 1 ) 2 is a simple formally real Jordan algebra of rank 3 and dim J c = 1 + n 2. The Jordan-von Neumann-Wigner classification implies that either dim J c = 1 or dim J c = 3d + 3, where d {1, 2, 4, 8}. Thus, n 2 = 0 or n 2 = 3d + 2. Using the obstruction implies the finiteness. n 1 1 ρ(n 1 + n 2 1) Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (21 of 26)
The Finiteness of Exceptional Hsiang Algebras Theorem B There exists finitely many isomorphy classes of exceptional Hsiang algebras. n 2 5 8 14 26 9 12 15 21 15 18 21 24 30 42 27 30 33 36 51 54 57 60 72 n 1 1 2 3 5 9 0 1 2 4 0 1 2 3 5 9 0 1 2 3 0 1 2 3 7 n 2 0 0 0 0 0 5 5 5 5 8 8 8 8 8 8 14 14 14 14 26 26 26 26 26 In the realizable cases (uncolored): If n 2 = 0 then u = 1 6 z, z2, z H 3 (A d ), d = 0, 1, 2, 4, 8. If n 1 = 0 then u(z) = 1 12 z2, 3 z z, z H 3(A d ), d = 2, 4, 8. If n 1 = 1 then u(z) = Re z, z 2, z H 3(A d ) C, d = 1, 2, 4, 8. If (n 1, n 2) = (4, 5) then u = 1 6 z, z2, z H 3(O) H 3(R) H 3(A d ) is the Jordan algebra of 3 3-hermitian matrices over the Hurwitz algebra A d Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (22 of 26)
Towards a finer classification The Tetrad Decomposition Let V be an exceptional Hsiang algebra, n 2 = 3d + 2. Then V = S 1 S 2 S 3 M 1 M 2 M 3, S α = S α S α, M α are nilpotent; each S α is a real division algebra isomorphic to A d ; Any vertex-adjacent triple (S α, S β, S γ) is a triality Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (23 of 26)
Some analytical corollaries Theorem F Let u be a nontrivial Hsiang cubic. Then u(x) = 0 the cubic trace identity holds: tr(d 2 u) 3 = 3λ(n 1 1)u, n 1 Z + n 2 = 1 (n + 1 3n1) Z+ 2 u(x) is exceptional Hsiang cubic iff n 2 2 and the quadratic trace identity holds tr(d 2 u) 2 = k x 2, k R Remark Peng Chia-Kuei and Xiao Liang (On the Classification of cubic minimal cones, J. Grad. School, Academ. Sinica, Vol.10, n.1, 1993): under assumption that u = 0 proved some particular results. Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (24 of 26)
N. Nadirashvili, V.G. Tkachev, and S. Vlăduţ, Nonlinear elliptic equations and nonassociative algebras, Mathematical Surveys and Monographs, vol. 200, American Mathematical Society, Providence, RI, 2014. V.G. Tkachev, A generalization of Cartan s theorem on isoparametric cubics, Proc. Amer. Math. Soc. 138 (2010), no. 8, 2889 2895., A Jordan algebra approach to the cubic eiconal equation, J. of Algebra 419 (2014), 34 51., On the non-vanishing property for real analytic solutions of the p-laplace equation, Proc. Amer. Math. Soc. 144 (2016)., Cubic minimal cones and nonassociative algebras, in preparation (2016). Vladimir G. Tkachev Hsiang algebras of cubic minimal cones Trondheim, November 4, 2016 (25 of 26)
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