HYPERSURFACES SATISFYING THE EQUATION Ax = Rx + b€¦ · (x, y) = xxyx + + xn+xyn+x - xn+2yn+2. Then {x £ W+X'x\(x, x) = -1} c R"+1,1 gives a natural isometric embedding of the

proceedings of theamerican mathematical societyVolume 120, Number 1, January 1994

HYPERSURFACES SATISFYING THE EQUATION Ax = Rx + b

JOONSANG PARK

(Communicated by Peter Li)

Abstract. We prove that a hypersurface in a space form or in Lorentzian spacewhose immersion x satisfies Ax = Rx + b is minimal or isoparametric. Inparticular, we locally classify such hypersurfaces which are not minimal.

1. Introduction

In [T] Takahashi proved that, if an isometric immersion x: Mn —► Rn+k sat-isfies Ax = -Xx , then M is either minimal in Rn+k (for X = 0) or minimal in§n+k~x(r) with r2 = n/X, where x = (xx,..., xn+k), Ax = (Axx,... , Axn+k),and A is the Laplacian on M given by the induced metric. Garay generalizedthis theorem for the hypersurfaces x: M" -* R"+1 satisfying Ax = Dx , whereD is a constant (n + 1) x (n + 1) diagonal matrix. He proved that such ahypersurface is either minimal or an open subset of a sphere or of a cylinder[G].

Let R"+1 ■' be Lorentzian space with the metric

(x, y) = xxyx + ■■■ + xn+xyn+x - xn+2yn+2.

Then {x £ W+X'x\(x, x) = -1} c R"+1,1 gives a natural isometric embeddingof the hyperbolic space H"+1 . So there exist n + 2 coordinate functions on animmersed hypersurface M in Mn+X or in S"+1 , i.e., x: M" -> H"+1 C Rn+1>1or x: Mn -* S"+1 c R"+2. In particular, we let Ax = (Axi, ... , Axn+2).

The purpose of this paper is to classify the isometric immersions of hypersur-faces in a simply connected space form Nn+X(c) or in Lorentzian space R"'1satisfying Ax = Rx + b, where R is a constant square matrix and b is a con-stant vector. This has been done in [AFL] for immersed surfaces in R2 •' orR3. Our main results are:

Theorem 1.1. Let x: M" -> R"+1 (or R"'1) be an isometric immersion withnondegenerate induced metric satisfying Ax = Rx + b, where R is a constant(n + 1) x (n + 1) matrix and b is a constant vector in R"+1 (or R">'). ThenM is minimal or isoparametric.

Received by the editors April 27, 1992.1991 Mathematics Subject Classification. Primary 57R42.Research supported in part by Global Analysis Research Center.

© 1993 American Mathematical Society0002-9939/93 $1.00 + $.25 per page

317

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318 JOONSANGPARK

Theorem 1.2. Let x: M" —> S"+1 (or Mn+X) be an isometric immersion satis-fying Ax = Rx, where R is a constant (n + 2) x (n + 2) matrix. Then M isminimal or isoparametric.

Theorem 1.3. In the above theorems, if M is isoparametric and not minimal,then M has at most two distinct principal curvatures. Moreover, it is an openpiece of one of the following, up to rigid motions:

(a) S"(r), Sk(r) xRn~k in Rn+X;(b) S"(r), Sk(rx)xS"-k(r2) in S"+1 ;(c) R", Mn(r), §"(/•), Sk(rx) xl"4(r2) in M"+1;(d) Sk(r) xR"-*-1'1, R"-kxMk(r), Rn~k x Sk(r) in Rn>x.

To prove these, we will show first that M has a constant mean curvature H,and then M will turn out to be isoparametric when H ^ 0. Isoparametric sub-manifold theory in space forms has been much developed recently [M, Ma, PT,W]. Munzner showed that the number of principal curvatures of isoparametrichypersurfaces in the sphere has to be g = 1, 2, 3, 4, or 6, and it is knownthat, when g = 1 or 2, the hypersurface must be a hypersphere or a product oftwo spheres.

This paper is organized as follows. In §2 we set up notation and review ba-sic facts about submanifold geometry in space forms and derive a differentialidentity relating the Laplacian of the mean curvature vector to the shape op-erator. This is needed for the proof of our main results. In §3 examples ofhypersurfaces satisfying Ax = Rx + b will be listed, providing the converse ofthe above theorems. In the last section, we prove the main theorems in eachambient space.

2. PreliminariesSuppose x: M" -> Rn+k is an isometric immersion. A local orthonormal

frame field ex, ... , en+k in R"+k is said to be adapted to M, if when restrictedto M, ex, ... , e„ are tangent to M. From now on, we shall use the followingindex convention:

1

HYPERSURFACES SATISFYING THE EQUATION Ax = Rx + b 319

The mean curvature vector H is then defined as

H = J2 Haea = tr II = ^ hiaiea.a i ,a

Now, let / be a C°° function on M. The Laplacian of / is defined by

A/ = trVV/ = trVrf/.Locally, it is defined as follows: Let

df=^2ficOj, Vdf=

320 JOONSANG PARK

As in the Euclidean case, it is easy to check that hijk is symmetric in i, j, kand Ax = H = Hen+X. Let / € C°°(M) and df = ^fito1. Then Vf, thegradient of /, and Vdf are given by

v/=YEi (vf > ee< = Y 6™C0™J )®eJ-Y hHhJkC°k ® en+\j \ m m I j ,/c

= - Y(hiJk(°k ® ei + hunjkU>k ® e„+i).j,k

Hence, Aen+X = - £, / %/£/ + h2jen+x . Now the theorem follows:

Y2{Hen+x)u = Y2H"en+\ + 2Hi(en+x)i + H(en+X)ui (

= AH • en+x - 2 £ hijH,ej - H I £ Hjej + Y hh ' en+l Iij \ j i.j J

since Hj = £i h»j = £, hm ■ D

3. ExamplesIf x: Mn —> Nn+X(c) is minimal, then it is well known (cf. [PT]) that

Ax = -cnx, and if x: Mn —> R" •' is minimal, then Ax = 0.



In the following, we denote by e„+x a unit normal vector on M, and we willgive examples of hypersurfaces that satisfy the differential equation Ax = Rx+bfor some R and b.3.1. Euclidean case, (a) M = {x £ R"+x: \\x - c\\2 = r2} , a standard sphere.Then en+x = (x - c)/r, and

Ax = --~7 -x + -*c.r2 r2

(b) Let M = {(y, z) £ Rk+X x R"~k+X: \\y - c\\2 = r2}, a cylinder. Thenen+\ = ((y-c)/r, 0), and

-=(-r DdHty3.2. Spherical case, (a) M = S"(sin0) c S"+1. Let x = y + v where y =(xx, ... , x„+i, 0) and v = (0, ... , 0, cosd). Then en+x = cot 0 • y - tan 0 • vand

f-ncsc29I 0\Ax={ o o)x-

(b) M = S*(cos0) x S"-/c(sinc/) c S"+1 . Let x = (x,, x2) £ Rk+X x R"~k ;then en+x = (- tan6 • xx, cot 8 - x2) and

. _(-k sec2 0/ 0 \*x-\ 0 -(n-k)csc26l)x-

3.3. Hyperbolic case, (a) M = S"(sinh(9) c H"+1. Let x = (y,cosh

322 JOONSANG PARK

(b) M = Rn~k x H*(r). Let x = (y, z); then en+x = (0, z/r),

A*=(o ki°/r2)X-(c) M = R"~k x §k(r). Let x = (y, z); then en+1 = (0, z/r),

AX={


are equal; hence, we get

{rtj = -Hhu - cndij,n,n+l=Hi(ri>n+2 = 0 forc^O).

To calculate rn+x A, from (A + cn)x = (R + cn)x + b = Hen+X , we have

R(Hen+x) = A(Hen+x),

and, from Theorem 2.1, we get

' Hrn+x>i = -2YljhijHj-HHi,(4-2) I Hrn+i,n+x= AH - H\\A\\2

. (Hrn+x>n+2 = cH2 for c = ±1).

On the other hand, from Rx = -cnx + Hen+X when c = ±1,

{rn+2,i = 0,

rn+2,n+x =H,rn+2,n+2 = -cn.

For the Lorentzian case, define rAB by R = YjAb yabCoa ®es ■ Then by a similarargument we obtain

(4.4) (rij = -eeJHhij,I ri,n+\ = Hi,

j Hrn+X, i = -2££i Y,j CjhjjHj - ee,////,,\//r„+1>n+1=A//-e//|M||2.

From (4.1) and (4.4), (r,7) = -H• A in N"+1(c),ot (eyru) = -H• A in Wn+xhjk for N = Nn+x(c),

£jrn+\,jhlk - £jrn+XJhjk = £rjtn+xhik - eri

324 JOONSANG PARK

Proposition 4.2. (T(VH), v)A(w) = (T(V//), w)A(v) Vv , w £ C°°(TM).Proof. Using Proposition 4.1, (4.1), (4.2), (4.4), and (4.5), we see that

for N = Nn+x(c),

\YhJiHi + HHj) hik = \ThuH, + ////,) hjk ,Vi,j,k;

for N = R"-X

(YzihHi + HHj) hik = \Y £ihdH, + HH-] hjk , Vi ,j,k. D

Proposition 4.3. M has constant mean curvature.Proof. To prove this, we will show that V//2(p) = 0 Vp 6 M. Suppose not.Then H(p)VH(p) ^ 0 for some p £ M.

Case 1. T(VH)(p) ^ 0. Then, by the nondegeneracy of the metric, thereexists a local tangent vector field v such that (T(VH) ,v) ^ 0. This impliesthat A has rank 1 on a neighborhood U of p by Proposition 4.2. We will geta contradiction in each ambient space case.

(i) N = Rn+X. Choose et so that{Aet = Xtei,

XX=H,X, = 0, i>\,

on U; then

(4.7) Oii,n+\ =XjCOi.

We will show that coXj = 0. Put dXj = Y,k Xikcok , cojj = Y,k ytjkcok .From Proposition 4.2 when i = k = 1, j > 1, we have Hj = 0. Hence,

XXj = Hj: = 0, i.e.,(4.8) Xij = 0 when i > 1, or i = 1 and / > 1.Take d of (4.7), and using the Codazzi and the structure equations,

dXj A cot = Y,(^J ~ ^»')


Hence the matrix of R with respect to eA is of the form

/ -H2 0 •• • 0 Hx \0 0 •■• 0 0

0 0 ••• 0 0V-3//i 0 ••• 0 ex(Hx)/H-H2)

by (4.1), (4.2), and \\A\\2 = H2. Hence, we have

(4.10) ex(Hx)/H-2H2 = cx,

(4.11) -Hex(Hx) + H4 + 3Hx2 = c2,

where the constants cx and c2 come from

det(/7 - i?) = 5I(-l)'c^"-'.i

Eliminate ex(Hx) in (4.10) and (4.11) to obtain

3Hx2-H4 = cxH2 + c2.

Differentiate this; then, since Hx ^ 0,

(4.12) 3ex(Hx)-2H3 = cxH.By (4.10) and (4.12), we conclude that H is a constant, which is a contradiction,

(ii) TV = Nn+X(c), c = ±1. Choose et so that

(Ae, = A,e,,X„ = H, on U.Xi = 0, i < n,

Then by Proposition 4.2, we have //, = 0 when i < n; hence, the matrix ofR + cnl with respect to eA is of the form

/0 ••• 0 \

/ -H2 Hn 0 \ ;0 -3Hn * cH

\ V 0 H 0)jhence, c//4 = -c$, where c3 is the coefficient of t3 in det((/? + cnl) - tl).We conclude that H is a constant, which is a contradiction.

(iii) N = R"'X . From the canonical forms of selfadjoint operators (see [O]),A is conjugate to the diagonal matrix

f° \0\ EH/

with respect to some orthonormal frame t?,. Hence, we have -co'n+l = Xjco',where A, = 0 for i < n , and X„ = eH . In a way similar to the Euclidean case,


326 JOONSANG PARK

we can prove that co" = 0 and AH = Ene„(H„). Hence, the matrix of R withrespect to eA is of the form

(° \0

-eH2 Hn\ -3s£„Hn £n£n(Hn)/H-EH2J

Again as in the Euclidean case, by solving a differential equation, we concludethat H is a constant, which is a contradiction.

Case 2. T(VH)(p) = 0. By Case 1, this holds on a neighborhood U of p .We will discuss the spherical and the Lorentzian cases only.

(i) For N = Sn+X . It implies that A(VH) = -HVH, i.e., -H is a principalcurvature. Therefore, we can choose e, so that

{A(et) = Xid,Xn = -H, onU.VH = Hnen,

Now R + nl is represented as the matrixl-HXx . 0 0 0\

: -HX2 : : :

0 . -HXn H„ 00 . //„ * H

\ 0 . OHO/with

n

H = 2_^Xi, X„ = -H.;'=1

This implies that -HXX, ... , -HXn_x are eigenvalues of the constant matrixR + nl and hence all of them are constants, so - Y%=\ HXi = -H(H - A„) =-2H2 is a constant, which is a contradiction.

(ii) For R"'1. If |V//| ^0,then

A={i -°eh) on(V//)x©R(V//)

where A' is selfadjoint. By choosing en = V///|V//|, R can be written as

f-H-A' $-£H2 Hn$.

Hence the eigenvalues of -H • A' are constants and so is H • trA' = H • trA -(-eH2) = 2eH2 , which is a contradiction.

If |V//| = 0 locally, then choose a pseudo-orthonormal frame ex, ... , e„such that \en-X\ = \en\ =0, (en-X, en) = 1, and e„ is parallel to V//. Then



where D„_k is an (n - fc)-diagonal matrix and

«-(-?■£)■ Hi-fi)-This gives the matrix for R

f-H-D„_k 0 0\0 -//-4 * .

\ 0 * */This implies that H • trDn_k = H • (tr A - tr A'k) = (k + \)eH2 is a constant,which is a contradiction.

Hence, we have

//(p)V//(p) = ±V//(p)2 = 0 VpeM,i.e., H is a constant. □

Proposition 4.4. If H ^ 0, then M is isoparametric.Proof. If H £ 0, then B = jj(R + cnl) is conjugate to (~0A°). Since Bis a constant matrix, ,4 is conjugate to a constant matrix, so M is isopara-metric. □

Now we prove the main theorems. All the isoparametric hypersurfaces inR"+1 are known to be hyperplanes, standard spheres, and circular cylinders [S],where the first one is minimal. For the hyperbolic case, Cartan [C] showedthat the examples in §2, (2.3) are the only possible cases. The isoparametrichypersurfaces in W •' are classified [Ma] and the examples in §2 are all ofthem, but in our situation, (a)-(c) are the only possible cases when H / 0. Forthe spherical case, we needTheorem 4.5. If x: M —> S"+1 satisfies Ax = Rx and H ^ 0, then the numberof principal curvatures is either 1 or 2.Proof. Let TM = Ex © • • • © Eg be the eigendecomposition corresponding tothe principal curvatures X,>, i = \ ,2, ... , g, and let E\ be the eigenspace ofB corresponding to A,. Then Et(p) C E\, Vp £ M. But we know the leaf ofEj is a sphere, which should belong to E\, so dim £, < dim E\. From

g gn = ^2dimE, < ^dim£- < n + 2 = dimR"+2,

;=1 /=1

we obtain g = 1 or 2. □

As mentioned before, the isoparametric hypersurfaces in §"+1 with g = 1or 2 are hyperspheres or products of two spheres. This completes the proofsof the theorems in § 1.

REFERENCES

[AFL] L. J. Alias, A. Ferrandez, and P. Lucas, Surfaces in the 3-dimensional space satisfyingAx = Ax + B , preprint.

[CI] E. Cartan, Sur les varietes de courbure constante dans I'espace euclidien ou non euclidien I,Bull. Soc. Math. France 47 (1919), 125-160.


328 JOONSANG PARK

[C2] _, Sur les varietes de courbure constante dans Tespace euclidien ou non euclidien II, Bull.Soc. Math. France 48 (1920), 132-208.

[G] O. J. Garay, An extension of Takahashi's Theorem, Geom. Dedicata 34 (1990), 105-112.[Ma] M. A. Magid, Lorentzian isoparametric hypersurfaces, Pacific J. Math. 118 (1985), 165-197.[M] H. F. Munzner, Isoparametrische Hyperfldchen in Sphdren I, Math. Ann. 251 (1980), 57-71.[O] B. O'Neill, Semi-Riemannian geometry with applications to relativity, Academic Press, New

York, 1983.[PT] R. S. Palais and C. L. Terng, Critical point theory and submanifold geometry, Lecture Notes

in Math., vol. 1353, Springer-Verlag, New York, 1988.[S] B. Segre, Famiglie di ipersuperifici isoparametriche neglie spazi euclidei ad un qualunque

numero di dimension, Atti Accad. Naz. Lincei Rend. CI. Sci. Fis. Mat. Natur. 27 (1938),203-207.

[T] T. Takahashi, Minimal immersions ofRiemannian manifolds, J. Math. Soc. Japan 18 (1966),380-385.

[W] B. L. Wu, Lorentzian isoparametric submanifolds, Ph.D. Thesis, Brandeis University, 1991.

Department of Mathematics, Brandeis University, Waltham, Massachusetts 02554-9110

Current address: Department of Mathematics, Dongguk University, Seoul 110-715, KoreaE-mail address: jparkQkrdguccl .bitnet


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HYPERSURFACES SATISFYING THE EQUATION Ax = Rx + b€¦ · (x, y) = xxyx + + xn+xyn+x - xn+2yn+2. Then {x £ W+X'x\(x, x) = -1} c R"+1,1 gives a natural isometric embedding of the