Explicit p-harmonic functions on the real Grassmannians

Elsa Ghandour; Sigmundur Gudmundsson

doi:10.1515/advgeom-2023-0015

Open Access Published by De Gruyter July 14, 2023

Explicit p-harmonic functions on the real Grassmannians

Elsa Ghandour and Sigmundur Gudmundsson

From the journal Advances in Geometry

https://doi.org/10.1515/advgeom-2023-0015

Abstract

We use the method of eigenfamilies to construct explicit complex-valued proper p-harmonic functions on the compact real Grassmannians. We also find proper p-harmonic functions on the real flag manifolds which do not descend onto any of the real Grassmannians.

Keywords: p-harmonic functions; symmetric spaces

MSC 2010: 53C35; 53C43; 58E20

1 Introduction

Mathematicians and physicists have been studying biharmonic functions for nearly two centuries. Applications have been found within physics for example in continuum mechanics, elasticity theory, as well as two-dimensional hydrodynamics problems involving Stokes flows of incompressible Newtonian fluids. Until just a few years ago, with only very few exceptions, the domains of all known explicit proper p-harmonic functions have been either surfaces or open subsets of flat Euclidean space. A recent development has changed this situation and can be traced at the regularly updated online bibliography [4], maintained by the second author.

A natural habitat for the study of complex-valued p-harmonic functions φ : (M, g) → ℂ, on Riemannian manifolds, is found by assuming that the domain is a symmetric space. These were classified by the pioneering work of Élie Cartan in the late 1920s. For this we refer to the standard work [11] by Helgason. The irreducible Riemannian symmetric spaces come in pairs each consisting of a compact space U/K and its non-compact dual G/K. In a recent article [9], the authors construct the first known explicit proper p-harmonic functions (p ≥ 2) for the compact cases

SO(n),SU(n),Sp(n)

of Type II, see also [5] and [7] for the special case when p = 2. For compact symmetric spaces of Type I, the authors of [8] deal with the cases of

SU(n)/SO(n),Sp(n)/U(n),SO(2n)/U(n),SU(2n)/Sp(n).

For complex-valued p-harmonic functions on symmetric spaces we have a duality principle first introduced for harmonic morphisms in [10] and later developed for p-harmonic functions in [5]. This means that a solution on the compact U/K induces, in a natural way, a solution on its non-compact dual G/K and vice versa. For this reason we discuss here only the compact cases.

It is the principal aim of this work to construct the first known explicit complex-valued proper p-harmonic functions on the real Grassmannians SO(m+n)/SO(m)×SO(n). Our method is inspired by the classical spherical harmonics on the standard round sphere S_n = SO(n + 1)/SO(n), see Remark 4.6.

2 Proper p-harmonic functions

In this section we describe a method for manufacturing complex-valued proper p-harmonic functions on Riemannian manifolds. This was recently introduced in [9].

Let (M, g) be an m-dimensional Riemannian manifold and T^ℂM be the complexification of the tangent bundle TM of M. We extend the metric g to a complex-bilinear form on T^ℂM. Then the gradient ∇φ of a complex-valued function φ : (M, g) → ℂ is a section of T^ℂM. In this situation, the well-known complex linear Laplace–Beltrami operator (alt. tension field) τ on (M, g) acts locally on φ as follows:

τ(φ)=div(∇φ)=∑i,j=1m1|g|∂∂xjgij|g|∂φ∂xi.

For two complex-valued functions φ,ψ:(M,g)→C we have the following well-known relation

(2.1) τ(φ⋅ψ)=τ(φ)⋅φ+2⋅κ(φ,ψ)+φ⋅τ(ψ),

where the complex bilinear conformality operator κ is given by κ(φ,ψ)=g(∇φ,∇ψ). Locally this satisfies

κ(φ,ψ)=∑i,j=1mgij⋅∂φ∂xi∂ψ∂xj.

Definition 2.1

([6]). Let (M, g) be a Riemennian manifold. Then a complex-valued function φ : M → ℂ is said to be an eigenfunction if it is eigen both with respect to the Laplace–Beltrami operator τ and the conformality operator κ, i.e. there exist complex numbers λ, μ ϵ ℂ such that

τ(φ)=λ⋅φ and κ(φ,φ)=μ⋅φ2.

A set E=φi:M→C∣i∈I of complex-valued functions is said to be an eigenfamily on M if there exist complex numbers λ, μ ϵ ℂ such that for all φ,ψ∈E we have

τ(φ)=λ⋅φ and κ(φ,ψ)=μ⋅φ⋅ψ.

In this work we are mainly interested in complex-valued proper p-harmonic functions. They are defined as follows.

Definition 2.2

Let (M, g) be a Riemannian manifold. For a positive integer p, the iterated Laplace–Beltrami operator τ^p is given by

τ0(φ)=φandτp(φ)=τ(τ(p−1)(φ)).

We say that a complex-valued function φ : (M, g) → ℂ is

p-harmonic if τ^p(φ) = 0, and
proper p-harmonic if τ^p(φ) = 0 and τ^(p−1)(φ) does not vanish identically.

Our construction of complex-valued proper p-harmonic functions, on the real Grassmannians, is based on the following method, recently introduced in [9].

Theorem 2.3

Let φ : (M, g) → ℂ be a complex-valued function on a Riemannian manifold and (λ, μ) ϵ ℂ² \ {0} be such that the tension field τ and the conformality operator κ satisfy

τ(φ)=λ⋅φandκ(φ,φ)=μ⋅φ2.

Then for any positive natural number p, the non-vanishing function Φ_p : W = {x ϵ M | φ(x) ∉ (−∞, 0]} → ℂ with

Φp(x)=c1⋅log(φ(x))p−1,ifμ=0,λ≠0c1⋅log(φ(x))2p−1+c2⋅log(φ(x))2p−2,ifμ≠0,λ=μc1⋅φ(x)1−λ/μlog(φ(x))p−1+c2⋅log(φ(x))p−1,ifμ≠0,λ≠μ

is a proper p-harmonic function. Here c₁, c₂ are complex coefficients, not both zero.

3 Lifting properties

We shall now present an interesting connection between the theory of complex-valued p-harmonic functions and the notion of harmonic morphisms. Readers not familiar with harmonic morphisms are advised to consult the standard text [2] and the regularly updated online bibliography [3].

Proposition 3.1

Let π:(Mˆ,gˆ)→(M,g) be a submersive harmonic morphism between Riemannian manifolds. Further let f : (M, g) → ℂ be a smooth function and fˆ:(Mˆ,gˆ)→C be the composition fˆ=f∘π. If λ:Mˆ→R+ is the dilation of π, then the tension fields τ and τˆ satisfy

τ(f)∘π=λ−2τˆ(fˆ)andτp(f)∘π=λ−2τˆ(λ−2τˆ(p−1)(fˆ))forallpositiveintegersp≥2.

Proof. The harmonic morphism π is a horizontally conformal, harmonic map. Hence the well-known composition law for the tension field gives

τˆ(fˆ)=τˆ(f∘π)=trace∇df(dπ,dπ)+df(τˆ(π))=λ2τ(f)∘π+df(τˆ(π))=λ2τ(f)∘π.

For the second statement, set h = τ(f) and hˆ=λ−2⋅τˆ(fˆ). Then hˆ=h∘π and it follows from the first step that

τˆ(λ−2τˆ(fˆ))=τˆ(hˆ)=λ2τ(h)∘π=λ2τ2(f)∘π,

or equivalently, τ2(f)∘π=λ−2τˆ(λ−2τ(fˆ)). The rest follows by induction. □

Let G be the special orthogonal group SO(m + n), with subgroup K = SO(m) × SO(n). Then the standard biinvariant Riemannian metric on G, induced by the Killing form, is Ad(K)-invariant and induces a Riemannian metric on the symmetric quotient space G/K. Moreover, the natural projection π : G → G/K is a Riemannian submersion with totally geodesic fibres, hence a harmonic morphism satisfying the conditions in Proposition 3.1.

4 Eigenfamilies on the real Grassmannians

The special orthogonal group SO(m + n) is the compact subgroup of the real general linear group GL_m_+n(ℝ) of invertible matrices satisfying

SO(m+n)={x∈GLm+n(R)∣x⋅xt=I and detx=1}.

Its standard representation on ℂ^m⁺ⁿ is denoted by

π:x↦x11⋯x1,m+n⋮⋱⋮xm+n,1⋯xm+n,m+n.

For this situation we have the following basic result from Lemma 4.1 of [6].

Lemma 4.1

For 1 ≤ j, k, α, β ≤ m+ n, let x_j : SO(m+ n) → ℝ be the real-valued matrix elements of the standard representation of SO(m + n). Then the following relations hold:

τˆ(xjα)=−(m+n−1)2⋅xjαandκˆ(xjα,xkβ)=−12⋅(xjβxkα−δjkδαβ).

For 1 ≤ j, α ≤ m+n, we now define the real-valued functions φˆjα:SO(m+n)→R on the special orthogonal group SO(m + n) by

φˆjα(x)=∑t=1mxjtxαt.

These functions are SO(m) × SO(n)-invariant and hence they induce functions on the compact quotient space SO(m+n)/SO(m)×SO(n), i.e. on the real Grassmannian G_m(ℝ^m⁺ⁿ) of m-dimensional oriented subspaces of ℝ^m⁺ⁿ.

Lemma 4.2

The tension field τˆ and the conformality operator κˆ on the special orthogonal group SO(m+n) satisfy

τˆ(φˆjα)=−(m+n)⋅φˆjα+δjα⋅mκˆ(φˆjα,φˆkβ)=−(φˆjβ⋅φˆkα+φˆjk⋅φˆαβ)+12(δjkφˆαβ+δαβφˆjk+δjβφˆkα+δkαφˆjβ).

Proof. The following calculations are based on the Equation (2.1) and the formulae in Lemma 4.1. For the tension field τˆ we have

τ ^ ( φ ^ j α ) = ∑ t = 1 m { τ ^ ( x j t ) ⋅ x α t + 2 ⋅ κ ^ ( x j t , x α t ) + x j t ⋅ τ ^ ( x α t ) } = − ( m + n − 1 ) ⋅ ∑ t = 1 m x j t x α t − ∑ t = 1 m ( x j t x α t − δ j α ) = − ( m + n ) ⋅ φ ^ j α + δ j α ⋅ m .

For the conformality operator κ we then obtain

κ ^ ( φ ^ j α , φ ^ k β ) = ∑ s , t = 1 m κ ^ ( x j s x α s , x k t x β t ) = ∑ s , t = 1 m { x j s x k t ⋅ κ ^ ( x α s , x β t ) + x j s x β t ⋅ κ ^ ( x α s , x k t ) + x α s x k t ⋅ κ ^ ( x j s , x β t ) + x α s x β t ⋅ κ ^ ( x j s , x k t ) } = − 1 2 ∑ s , t = 1 m { x j s x k t ⋅ ( x α t x β s − δ α β δ s t ) + x j s x β t ⋅ ( x α t x k s − δ α k δ s t ) = + x α s x k t ⋅ ( x j t x β s − δ j β δ s t ) + x α s x β t ⋅ ( x j t x k s − δ j k δ s t ) } = − ( φ ^ j β ⋅ φ ^ k α + φ ^ j k ⋅ φ ^ β α ) + 1 2 ⋅ ( δ j k φ ^ α β + δ α β φ ^ j k + δ j β φ ^ k α + δ k α φ ^ j β ) .

Theorem 4.3

Let A be a complex symmetric (m + n) × (m + n)-matrix such that A² = 0. Then the SO(m) × SO(n)-invariant function ΦˆA:SO(m+n)→C given by

ΦˆA(x)=∑j,α=1m+najα⋅φˆjα(x)

induces an eigenfunction Φ_A : SO(m + n)/SO(m) × SO(n) → ℂ on the real Grassmannian with

τ(ΦA)=−(m+n)⋅ΦAandκ(ΦA,ΦA)=−2⋅ΦA2

if rank A = 1 and trace A = 0.

Proof. It is an immediate consequence of Lemma 4.2 and the fact that A is traceless that the tension field τˆ satisfies

τˆ(ΦˆA)=−(m+n)⋅ΦˆA+m⋅traceA=−(m+n)⋅ΦˆA.

For the conformality operator κ we have

2 ⋅ Φ ^ A 2 + κ ^ ( Φ ^ A , Φ ^ A ) = 2 ⋅ Φ ^ A 2 + ∑ j , α , k , β = 1 m + n κ ^ ( a j α φ ^ j α , a k β φ ^ k β ) = 2 ⋅ Φ ^ A 2 + ∑ j , α , k , β = 1 m + n a j α a k β ⋅ κ ^ ( φ ^ j α , φ ^ k β ) = 2 ⋅ Φ ^ A 2 − ∑ j , α , k , β = 1 m + n a j α a k β { φ ^ j β ⋅ φ ^ k α + φ ^ j k ⋅ φ ^ β α } = + 1 2 ∑ j , α , k , β = 1 m + n a j α a k β { δ j k φ ^ α β + δ α β φ ^ j k + δ j β φ ^ k α + δ k α φ ^ j β }

= 2 ⋅ Φ ^ A 2 − ∑ j , α , k β = 1 m + n { a j β a k α + a j k a α β } ⋅ φ ^ j α φ ^ k β = + 1 2 ∑ j , α , k , β = 1 m + n a j α a k β ( δ k s φ ^ j r + δ k r φ ^ j s + δ j s φ ^ k r + δ j r φ ^ k s ) = ∑ j , α , k , β = 1 m + n { 2 a j α a k β − a j β a k α − a j k a α β } ⋅ φ ^ j α φ ^ k β + 2 ∑ j , α , t = 1 m + n a j t a α t φ ^ j α = ∑ j , α , k , β = 1 m + n { det a j α a j β a k α a k β + det a j α a j k a β α a β k } ⋅ φ ^ j α φ ^ k β + 2 ∑ j , α = 1 m + n ( a j , a α ) ⋅ φ ^ j α = 0 ,

since A² = 0 and rankA = 1. □

Proposition 4.4

Let A be a complex (m + n) × (m + n) matrix such that

Ωjk(α,β)=detajαajβakαakβ+detajαajkaβαaβk=0,

for all 1 ≤ j, k, α, β ≤ m + n. Then A² = A - trace A and rank A = 1.

Proof. The first statement is an immediate consequence of the relation

∑α=1m+nΩjk(α,α)=(aj,ak)−ajk⋅traceA.

The second statement follows from

Ωjk(α,β)−Ωjk(β,α)=3{ajαakβ−ajβakα}=3⋅detajαajβakαakβ.

In Example 4.5, we now construct matrices satisfying the conditions in Theorem 4.3 and hence manufacture a multi-dimensional family of eigenfunctions on the real Grassmannian SO(m + n)/SO(m) × SO(n).

Example 4.5

Let p = (p₁, . . . , p_m_+n) ϵ ℂ^m⁺ⁿ be a non-zero isotropic element, i.e.

p12+p22+…+pm+n2=0.

Then the complex (m + n) × (m + n) matrix A = p^t - p with a_jk = p_jp_k satisfies the conditions A² = 0, trace A = 0 and rank A = 1. Furthermore, the SO(m) × SO(n)-invariant function Φˆp:SO(m+n)→C with

Φˆp(x)=∑j,α=1m+npjpk⋅(∑t=1mxjtxkt)

induces an eigenfunction Φ_p : SO(m + n)/SO(m) × SO(n) → ℂ on the quotient space with

τ(Φp)=−(m+n)⋅Φpandκ(Φp,Φp)=−2⋅Φp2.

This provides a complex (m + n − 1)-dimensional family of eigenfunctions on the real Grassmannian manifold SO(m + n)/SO(m) × SO(n).

Next we explain how our construction method is inspired by the classical theory of spherical harmonics on Sⁿ, as the unit sphere in the Euclidean ℝⁿ⁺¹. For this see the excellent text [1].

Remark 4.6

For m = 1, we identify the first column of the generic matrix element

x11⋯x1,n+1⋮⋱⋮xn+1,1⋯xn+1,n+1

in SO(n + 1) with x = (x₁, x₂, . . . , x_n₊₁) ϵ ℝⁿ⁺¹. Then the linear space Hn+12 of second order harmonic polynomials in the coordinates of x ϵ ℝⁿ⁺¹ is generated by the elements

x12−x22,x22−x32,…,xn2−xn+12,x1x2,x1x3,…,xn−1xn+1,xnxn+1,

forming a basis ℬ for Hn+12 . Their restrictions to the unit sphere Sⁿ are eigenfunctions of the Laplace–Beltrami operator, all of the same eigenvalue.

By assuming, in Theorem 4.3, that the matrix A is traceless we see that the SO(n)-invariant function ΦˆA: SO(n+1)→C given by

ΦˆA(x)=∑j,α=1n+1ajα⋅φˆjα(x)

is a linear combination of the basis elements in ℬ. If rankA = 1 and trace A = 0, then these functions are eigen with respect to the conformality operator κ.

5 Eigenfunctions on the real flag manifolds

The standard Riemannian metric on the special orthogonal group SO(n) induces a natural metric on the real homogeneous flag manifolds

F(n1,…,nt)=SO(n)/SO(n1)×SO(n2)×…×SO(nt),

where n = n₁ + n₂ + ⋯ + n_t . For this we have the Riemannian fibrations

SO(n)→F(n1,…,nt)→Gnk(Rn).

Let us now write the generic element x ϵ SO(n) in the form

x=[x1|x2|⋯xt],

where each x_k is an n×n_k submatrix of x. Following Theorem 4.3, we can now, for each block, construct a family Eˆk of SO(n_k)invariant complex-valued eigenfunctions on the special orthogonal group SO(n) such that for all φˆ,ψˆ∈Eˆk

τ(φˆ)=λk⋅φˆandκ(φˆ,ψˆ)=μk⋅φˆ⋅ψˆ,

with λ_k = −n and μ_k = −2. We denote by φˆk the generic element in Eˆk and then, according to Theorem2.3, each function

Φˆp,k(x)=c1,k⋅φˆk(x)1−λk/μklog(φˆk(x))p−1+c2,k⋅log(φˆk(x))p−1

is proper p-harmonic on an open and dense subset of SO(n). The sum

Φˆp=∑k=1tΦˆp,k

constitutes a multi-dimensional family Fˆp of SO(n₁)× ⋯ × SO(n_t)-invariant proper p-harmonic functions on an open dense subset of SO(n). Furthermore, each element Φˆp∈Fˆp induces a proper p-harmonic function Φp∗ defined on an open and dense subset of the real flag manifold

F(n1,…,nt)=SO(n)/SO(n1)×…×SO(nt),

which does not descend onto any of the real Grassmannians if t ≥ 3.

Acknowledgements

The authors would like to thank Fran Burstall and Adam Lindström for useful discussions on this work. The first author would like to thank the Department of Mathematics at Lund University for its great hospitality during her time there as a postdoc.

Table 1

Eigenfunctions on classical irreducible compact Riemannian symmetric spaces.

U/K	λ	μ	Eigenfunctions
SO(n)	−(n−1)2	−12	see [6]
SU(n)	−n2−1n	−n−1n	see [9]
Sp(n)	−2n+12	−12	see [5]
SU(n)/SO(n)	−2n2+n−2n	−4(n−1)n	see [8]
Sp(n)/U(n)	−2(n + 1)	−2	see [8]
SO(2n)/U(n)	−2(n − 1)	−1	see [8]
SU(2n)/Sp(n)	−22n2−n−1n	−2(n−1)n	see [8]
SO(m + n)/SO(m) × SO(n)	−(m + n)	−2	Theorem 4.3

Communicated by: F. Duzaar

References

[1] S. Axler, P. Bourdon, W. Ramey, Harmonic function theory. Springer 2001. MR1805196 Zbl 0959.3100110.1007/978-1-4757-8137-3Search in Google Scholar

[2] P. Baird, J. C. Wood, Harmonic morphisms between Riemannian manifolds, volume 29 of London Mathematical Society Monographs. New Series. Oxford Univ. Press 2003. MR2044031 Zbl 1055.5304910.1093/acprof:oso/9780198503620.001.0001Search in Google Scholar

[3] S. Gudmundsson, The Bibliography of Harmonic Morphisms. www.matematik.lu.se/matematiklu/personal/sigma/harmonic/bibliography.html.Search in Google Scholar

[4] S. Gudmundsson, The Bibliography of p-Harmonic Functions. www.matematik.lu.se/matematiklu/personal/sigma/harmonic/p-bibliography.html.Search in Google Scholar

[5] S. Gudmundsson, S. Montaldo, A. Ratto, Biharmonic functions on the classical compact simple Lie groups. J. Geom. Anal. 28 (2018), 1525–1547. MR3790510 Zbl 1414.5801110.1007/s12220-017-9877-1Search in Google Scholar

[6] S. Gudmundsson, A. Sakovich, Harmonic morphisms from the classical compact semisimple Lie groups. Ann. Global Anal. Geom. 33 (2008), 343–356. MR2395191 Zbl 1176.5801110.1007/s10455-007-9090-8Search in Google Scholar

[7] S. Gudmundsson, A. Siffert, New biharmonic functions on the compact Lie groups SO(n), SU(n), Sp(n). J. Geom. Anal. 31 (2021), 250–281. MR4203645 Zbl 1460.3102210.1007/s12220-019-00259-3Search in Google Scholar

[8] S. Gudmundsson, A. Siffert, M. Sobak, Explicit proper p-harmonic functions on the Riemannian symmetric spaces SU(n)/SO(n), Sp(n)/U(n), SO(2n)/U(n), SU(2n)/Sp(n). J. Geom. Anal. 32 (2022), Paper No. 147, 16 pages. MR4382667 Zbl 1486.3102010.1007/s12220-022-00885-4Search in Google Scholar

[9] S. Gudmundsson, M. Sobak, Proper r-harmonic functions from Riemannian manifolds. Ann. Global Anal. Geom. 57 (2020), 217–223. MR4057458 Zbl 1439.3100810.1007/s10455-019-09696-3Search in Google Scholar

[10] S. Gudmundsson, M. Svensson, Harmonic morphisms from the Grassmannians and their non-compact duals. Ann. Global Anal. Geom. 30 (2006), 313–333. MR2265318 Zbl 1103.5800710.1007/s10455-006-9029-5Search in Google Scholar

[11] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, volume 80 of Pure and Applied Mathematics. Academic Press 1978. MR514561 Zbl 0451.53038Search in Google Scholar

Received: 2022-03-15

Revised: 2022-09-01

Published Online: 2023-07-14

Published in Print: 2023-08-28

This work is licensed under the Creative Commons Attribution 4.0 International License.

Explicit p-harmonic functions on the real Grassmannians

Abstract

1 Introduction

2 Proper p-harmonic functions

Definition 2.1

Definition 2.2

Theorem 2.3

3 Lifting properties

Proposition 3.1

4 Eigenfamilies on the real Grassmannians

Lemma 4.1

Lemma 4.2

Theorem 4.3

Proposition 4.4

Example 4.5

Remark 4.6

5 Eigenfunctions on the real flag manifolds

Acknowledgements

References

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