RIGIDITY PROPERTIES OF CONVEXHYPERSURFACES

XI SISI SHEN - SUPERVISED BY PROF. PENGFEI GUANMATHEMATICS AND STATISTICS

PROBLEMTwo surfaces are considered congruent if theydiffer by a rigid motion such as a translationor rotation. For 2-dimensional surfaces, HeinzHopf has shown that two isometric convex sur-faces must be congruent. Does this result stillhold on surfaces in higher dimensions?

The Main Theorem Any two convex hyper-surfaces that are isometric must be congruent.

NOTIONS OF CURVATURECurvature for a curve can be thought of as ameasure of how much that curve deviates froma straight line at a given point.

The principal curvatures of a surface in R2 arethe min/max curvatures obtained by curves ofthe normal sections.

Curvature can also be expressed through theGauss map, denoted N , which is a map from asurface to the unit sphere. The differential of theGauss map, dN , is an nxn matrix whose eigen-values represent the principal curvatures.

Curvature R2 Rn

Principalκ1, κ2 =

eigenvalues of dNmap (2x2)

κ1, . . . , κn =eigenvalues of dN

map (nxn)

Gauss,K invariant underisometries

not invariant underisometries

Scalar,R

equals Gausscurvature, thus,invariant under

isometries

invariant underisometries

Scalar curvature is defined for any n-dimensional hypersurface A as:

R(A) =∑n

i=2

∑i−1j=1 κiκj

Polarized curvature is defined for two n-dimensional hypersurfaces A,B, as:

R′(A,B) = 12∑n

i,j=1,i6=j κai κ

bj

PROOF OF THE MAIN THEOREMLet us denote the two surfaces by M and M*. We may assume that the intersection of the interiorsof M and M* is non-empty and take point q in this intersection. Let p and p∗ represent the supportfunctions of M and M* respectively. Because the hypersurfaces are isometric, we may parametrizethem both on a common hypersurface, S. Our goal will be to show that the following integral is zero.∫

S

(R−R′)(p+ p∗)dV = 0.

Since p and p∗ were chosen from a point in the intersection of the interiors of M and M* which areeach convex, they are each strictly positive, thus, p + p∗> 0. By Lemma 1, R − R′ is non-positive,thus in showing that the integral is zero, we may conclude that R−R′ is identically zero. From there,Garding’s inequality tells us that κMi = κ

M∗i ∀1 ≤ i ≤ n. This means that M and M∗ share the same

principal curvatures, thus, by Lemma 3, they must be congruent.

To show that the integral is zero, let us expand it to obtain∫S

(Rp−R′p+Rp ∗ −R′p∗)dV

Using Chern’s integral formula, we have∫S

(P0,2p− P1,1p∗)dV =∫S0

(2

n(n− 1)Rp− 2

n(n− 1)R(M,M ′)p∗)dV = 0

∫S

(2

n(n− 1)Rp− 2

n(n− 1)R′p∗)dV = 0 =⇒

∫S0

(Rp−R′p∗)dV = 0

It follows by symmetry, Lemma 2 and the fact that R(M) = R(M∗) that∫S

(R ∗ p ∗ −R′p)dV =∫S0

(Rp ∗ −R′p)dV = 0

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REFERENCES[1] Chern, Shiing-Shen; Integral Formulas for Hypersurfaces in Euclidean Space and Their Applications to Uniqueness Theorems In Journal of Mathematics and Mechanics, Vol 8, No. 6 (1959)[2] Hopf, Heinz; Differential Geometry in the Large In Lecture Notes in Mathematics, Vol. 1000 (1989)[3] Lars, Garding; An Inequality for Hyperbolic Polynomials In Journal of Mathematics and Mechanics, Vol. 8, No. 6 (1959)[4] Do Carmo, Manfredo; Differential Geometry of Curves and Surfaces In Prentice-Hall, Inc., Upper Saddle River, New Jersey (1976)

GENERALIZATIONSIn the proof, convexity is used only to justify thatthe surface has positive support function at allpoints. The support function, p, with normalmap, N, centered at a point Z, at a point x, isgiven by p(x) = x ·N(x).

For star-shaped hypersurfaces, there exists apoint Z such that the support function p is pos-itive for all points x on the hypersurface. Thus,the result may be generalized, with a few minoradjustments, to all star-shaped hypersurfaces.

LEMMASChern’s integral formula for compact hypersur-faces: ∫

S0

(P0,`p− P1,`−1p∗)dV = 0

It can be shown that in the case of ` = 2, thefollowing hold:

P0,2 =2

n(n− 1)R

P1,1 =2

n(n− 1)R′

Lemma 1 Given two isometric hypersurfacesM and M* of dimension n, the difference be-tween the scalar curvature R(M) = R(M∗)and the polarized curvature R′(M,M∗) is non-positive, i.e. R−R′≤ 0 for all dimensions n.

Lemma 2 The polarized scalar curvature issymmetric about its two entries:

R′(A,B) = R′(B,A)

Lemma 3 If two hypersurfaces share the sameprincipal curvatures, then they must differ by arigid motion.