Minimal Submanifolds of Higher Codimension
Yuanlong Xin
Institute of Mathematics, Fudan University, Shanghai 200433, China
and Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education
e-mail : [email protected]
(2000 Mathematics Subject Classification : 49Q05, 53A07, 53A10. )
1 Introduction
Minimal surfaces were introduced by Lagrange in 1762. Using the calculus of variations he derived the following equation
(1 +fy2)fxx−2fxfyfxy+ (1 +fx2)fyy = 0.
Belgian physicist J. Plateau in 1847 proposed Plateau problem. The problem is reduced to find a minimal surface with a boundary which is the given Jordan curve.
In 1930 this problem was settled by Douglas and Rado . We also have the following famous theorem.
Theorem 1.1. (Bernstein, 1915)[1] The entire minimal graph in R3 has to be a plane.
The minimal hypersurface equation inRn+1is (1 +|∇f|2)X
i
∂2f
∂x2i −X
i,j
∂f
∂xi
∂f
∂xj
∂2f
∂xi∂xj
= 0, where f is a function defined on a domain in Rn.
The higher dimensional version of Bernstein’s theorem was settled in 1960’s through the efforts of Federer, Fleming, de Giorgi, Almgren, Simons and Bombieri- de Giorgi-Giusti. The answer is that the equation has only linear entire solutions forn≤7 and there are non-trivial entire solution forn >7.There is a weak version of the Bernstein theorems as follows.
Theorem 1.2. (J. Moser, 1961)[9] Letz=f(x1,· · ·, xm)define an entire minimal graph in Rm+1. If |∇f| ≤β < ∞, then f is a linear function and its graph is a hyperplane.
The research was partially supported by NSFC and SFME.
59
This theorem was improved by geometric approach.
Theorem 1.3. (J.C.C.Nitscher and Ecker - Huisken, 1990) [10][2] An entire smooth solution to (2) satisfying
|∇f(x)|=o(p
|x|2+|f(x)|2) is an affine linear function.
Remark 1.1. Heinze (1952) [4] considered the minimal graph defined over a disc DR ⊂ R2 and gave curvature estimates. The classical Bernstein theorem can be obtained by lettingR→+∞in his curvature estimates. Schoen-Simon-Yau (1975) [12] found new curvature estimates for stable minimal hypersurfaces which reproved the Bernstein’s theorem for dimension up to 5.
2 Higher Codimension
LetM →M¯ be ann−submanifold in (m+n)−dimensional Riemannian mani- fold ¯M with the second fundamental formBwhich can be viewed as a cross-section of the vector bundle Hom(2T M, N M) overM, where T M and N M denote the tangent bundle and the normal bundle along M, respectively. A connection on Hom(2T M, N M) is induced from those ofT M andN M naturally.
Taking the trace of B gives the mean curvature vectorH of M in ¯M, a cross- section of the normal bundle [14].
M is called a minimal submanifold ifH ≡0.
There are many interesting examples of minimal submanifolds of higher codi- mension:
• The Veronese surface inS4.
• Any holomorphic curve inC2 is a minimal surface of codimension 2.
• Any complex submanifold in K¨ahler manifold is a minimal submanifold of codi- mension at least 2.
• Special Lagrangian submanifolds in a K¨ahler manifold are minimal submanifolds of higher codimension [5].
• Associate submanifolds, coassociate submanifolds · · ·[5] are minimal submani- folds of higher codimension.
Let Ω ⊂ Rm be an open domain. Let f : Ω → Rn be a smooth map. If it satisfies a system of nonlinear equations
X
i,j
gij ∂2f
∂xi∂xj = 0, where gij =δij+P
s
∂fs
∂xi
∂fs
∂xj,(gij) = (gij)−1 andg = det(gij), then its graph is a minimal submanifoldM inRm+n.
If Ω is bounded and convex, we can consider Dirichlet problem for the above minimal surface system and the given boundary data on∂Ω.
Minimal hypersurfaces possess many beautiful properties and we have quite deep understanding for the issue of minimal hypersurfaces in many aspects, whereas minimal submanifolds of higher codimension are more complicated. Lawson- Osserman’s paper shows several important different phenomena in higher codimen- sion[8]:
•The solutions to the Dirichlet problem for minimal surface system on the unit ball in R2of codimension 2 are not unique and one of the solutions is unstable.
• If the boundary data is sufficiently small the Dirichlet problem for minimal surface system is always solvable. Whereas there are examples of nonexistence in dimension
≥4.
• There exist non-parametric minimal cones, in particular, the exists a 4−dimensional minimal cone inR7.
3 On Lawson-Osserman’s Problem
Moser’s result has been partially generalized to certain cases in higher codimen- sion.
1. dimension 2 by Chern-Osserman, 1967;
2. dimension 3 by L. Barbosa, 1979.
3. Lawson-Osserman found [8] counter example: a minimal 4−submanifold in R7. They also raised a question for finding the ”best” constant possible in the theorem. It is related to Simons’ problem on extrinsic rigidity of minimal submanifolds in the sphere and the regularity problem for a system of non- linear PDE.
In general, we have following results.
Theorem 3.1. (Hildebrandt-Jost-Widmen, 1980)[6] Letzα=fα(x), α= 1,· · · , m, x= (x1,· · ·, xn)∈ Rn be theC2 solution to the system of minimal surface equations.
Let there exist β, where
β <cos−s π
2√ s K
, K=
1 if s= 1 2 if s≥2
, s= min(m, n)
such that for anyx∈Rn,
∆f(x) ={det(δij+fxsi(x)fxsj(x))}12 ≤β,
then f1,· · ·, fm are affine linear functions on Rn, whose graph is affinen−plane in Rm+n.
Theorem 3.2. (Jost - Xin, 1999)[7] Let zs = fs(x), s = 1,· · · , m, x = (x1,· · ·, xn) ∈ Rn be smooth functions whose graph defines a submanifold with parallel mean curvature in Rm+n. Let there exist
β <
2, m≥2
∞, m= 1 such that for anyx∈Rn,
∆f(x) ={det(δij+fxsi(x)fxsj(x))}12 ≤β,
thenf1,· · · , fmare linear functions onRn,whose graph is affinen−plane inRm+n. Our result improved the previous results of Hildebrandt-Jost-Widman. Its proof also uses their Liouville type theorem for certain harmonic maps, the key point is to find a larger geodesic convex set in Grassmannian manifolds. It is interesting in its own right.
For the flat normal bundle situation we can have better results.
Theorem 3.3. (Smoczyk-Wang-Xin, 2006)[11] Let M be a minimal submanifold inRm+n of codimension mwith flat normal bundle and positivew-function. If for q∈h
0,q
2 n
,
lim
R→∞R−(2q+4)vol(DR) = 0, thenM is flat.
Theorem 3.4. (Xin, 2005 [13]; Smoczyk-Wang-Xin, 2006 [11]) LetM = (x, f(x)) be an entire minimal graph given by p functions fα(x1,· · · , xn), p ≥2 with flat normal bundle. If
(det(δij+fiαfjα))12 =o(p
|x|2+|f(x)|2), thenfα are affine linear functions.
For the curved normal bundle, situation is more complicate. Using the knowl- edge of the Grassmannian manifolds we can define some auxiliary functions to over- come difficulties. We can carry out both Schoen-Siomon-Yau curvature estimates and Ecker-Huisken curvature estimates in higher codimension.
The Schoen-Simon-Yau type estimates give us the following Bernstein type the- orem.
Theorem 3.5. (Xin-Yang)[17] Let M be a complete minimal n-dimensional sub- manifold in Rn+m (n ≤ 6). If the Gauss image of M is contained in an open geodesic ball ofGn,mcentered atP0 and of radius
√ 2
4 π, thenM has to be an affine linear subspace.
Theorem 3.6. (Xin-Yang)[18] Let M = (x, f(x)) be an n-dimensional minimal graph given bym functionsfα(x1,· · ·, xn)withm≥2, n≤4. If
∆f =
"
det δij+X
α
∂fα
∂xi
∂fα
∂xj
!#12
<2
then fα has to be affine linear functions representing an affine n-plane.
The Ecker-Huisken type estimates yield
Theorem 3.7. (Xin-Yang)[17] Let M be a complete minimal n-dimensional sub- manifold in Rn+m. If the Gauss image of M is contained in an open geodesic ball of Gn,mcentered at P0 and of radius
√2 4 π, and
√2
4 π−ρ◦γ−1
has growth
(3.1)
√2
4 π−ρ◦γ−1
=o(R),
whereρdenotes the distance onGn,mfromP0andRis the Euclidean distance from any point in M. Then M has to be an affine linear subspace.
Theorem 3.8. (Xin-Yang)[18] Let M = (x, f(x)) be an n-dimensional minimal graph given bym functionsfα(x1,· · ·, xn)withm≥2. If
∆f =
"
det δij+X
α
∂fα
∂xi
∂fα
∂xj
!#12
<2, and
(3.2) (2−∆f)−1=o(R43),
whereR2=|x|2+|f|2. Thenfα has to be affine linear functions and henceM has to be an affine linear subspace.
4 Bochner Type Formula
Now, we consider a minimal n−submanifold M in Rm+n, m≥ 2 with second fundamental formB. We have [17]
(4.1) ∆|B|2≥ −3|B|4+ 2|∇B|2.
We need a Kato-type inequality in order to use the formula (4.1). Namely, we would estimate|∇B|2in terms of|∇|B||2.Schoen-Simon-Yau did such an estimate for codimension m= 1. For any mwith flat normal bundle their technique is also applicable. The following lemma is the refined version of their estimate for any codimensionm.
Lemma 4.1. [17]
(4.2) |∇B|2≥ 1 + 2
mn
∇|B|
2.
Remark 4.1. Recently, we obtained a more general inequality which also improved the above inequality in minimal cases [16].
Hence it follows from (4.1) and (4.2) that
(4.3) ∆|B|2≥2 1 + 2
mn
∇|B|
2−3|B|4.
This inequality is used both for Schoen-Simon-Yau type estimates and Ecker- Huisken type estimates for minimal submanifolds in higher codimension.
5 Two Auxiliary Functions via Gauss Map
We consider smooth functions on an open geodesic ball B√2 4
(P0) ⊂ Gm,n of radius
√2
4 πand centered atP0. Those are useful for our curvature estimates later.
Let
u= cos(√
2ρ), h= sec2(√ 2ρ) whereρis the distance function fromP0 inGm,n.
By the standard Hessian comparison theorem we have
(5.1) Hess(ρ)≥√
2 cot(√
2ρ)(g−dρ⊗dρ), Then, we have
(5.2) Hess(u)≤ −2ug.
The composition functionh1=u◦γ ofuwith the Gauss mapγ defines a function onM. Using composition formula, we have
(5.3) ∆h1≤ −2|B|2h1.
We also have
Hess(h)≥4h g+3
2h−1dh⊗dh
The composition functionh2=h◦γofhwith the Gauss mapγdefines a function onM. Using composition formula, we have
(5.4) ∆h2≥4 h2|B|2+3
2h−12 |∇h2|2,
One of the ingredients in Schoen-Simon-Yau estimates is Stability inequality.
With the help ofh1we obtain a strong stability inequality.
Lemma 5.1. [17] Let M be ann-dimensional minimal submanifold of Rn+m (M needs not be complete), if the Gauss image of M is contained in an open geodesic ball of radius
√ 2
4 πinGn,m, then we have (5.5)
Z
M
|∇φ|2∗1≥2 Z
M
|B|2φ2∗1 for any function φwith compact supportD⊂M.
The above auxiliary functions are in terms of intrinsic distance function in Grassmannian manifolds. We can also define similar auxiliary functions in terms of the extrinsic functions of Grassmannian manifolds as submanifolds via Plu¨cker imbedding [18].
ForP0 ∈Gn,m, which is expressed by a unitn−vector ε1∧ · · · ∧εn. For any P ∈Gn,m, expressed by ann−vectore1∧ · · · ∧en, we define an important function on Gn,m: hight function for Gm,n under Pl¨ucker embedding into the Euclidean space
wdef.= hP, P0i=he1∧ · · · ∧en, ε1∧ · · · ∧εni= detW, where W = (hei, εji).The Jordan angles betweenP andP0are defined by
θα= arccos(λα),
where λα ≥ 0 and λ2α are the eigenvalues of the symmetric matrix WTW. The distance betweenP0 andP in canonical Riemannian metric onGm,n is
d(P0, P) = qX
θ2α. Denote
U ={P ∈Gn,m:w(P)>0}.
OnU we can define
v=w−1=Y
α
secθα. For any real numberalet
Va ={P ∈Gn,m, v(P)< a}.
We have quite accurate estimates
Theorem 5.1. [18]
(5.6) Hess(v)≥v(2−v)g+ v−1
sv(v2s −1)+s+ 1 sv
dv⊗dv
onV2, whereg is the metric tensor onGn,mands=min(n, m).
6 Curvature Estimates
Let r be a function on M with |∇r| ≤ 1. For any R ∈ [0, R0], where R0 = supMr, suppose
MR={x∈M, r≤R}
is compact.
• Schoen-Simon-Yau type estimates:
From (4.3) and (5.5) we have
Theorem 6.1. (Xin-Yang)[17] Let M be ann-dimensional minimal submanifolds ofRn+m. If the Gauss image ofMR is contained in an open geodesic ball of radius
√2
4 π inGn,m, then we have theLp-estimate
(6.1)
|B|
Lp(M
θR)≤C(n, m, p)(1−θ)−1R−1Vol(MR)1p for any θ∈(0,1) andp∈h
4,4 +23+43q
1 +mn6 .
Theorem 6.2. (Xin-Yang) [18] LetM be an n-dimensional minimal submanifold of Rn+m. If the Gauss image ofMR is contained in {P ∈U ⊂Gn,m:v(P)<2}, then we have the estimate
(6.2)
|B|
Ls(M
θR)≤C(n, m, s)(1−θ)−1R−1Vol(MR)1s for arbitraryθ∈(0,1) and
s∈
4,4 + 4 3p+2
3 r
3 +2 p
6 mn+2
p
.
• With the help of the auxiliary functionh2 we can obtain subhamornic functions in the minimal submanifolds which enable us to carry out the Ecker-Huisken type estimates.
Theorem 6.3. (Xin-Yang)[17] LetM be ann-dimensional minimal submanifolds of Rn+m. If the Gauss image ofMR is contained in an open geodesic ball of radius
√ 2
4 πinGn,m, then there exists C1=C1(n, m),such that
(6.3)
|B|2h2 Lp(M
θR)≤C2(p)(1−θ)−2R−2 h2
Lp(M
R)
wheneverp≥C1 andθ∈(0,1).
LetBR(x)⊂Rm+nbe a ball of radiusRand centered atx∈M. Its restriction onM is denoted by
DR(x) =BR(x)∩M.
The aboveLpestimate and mean value inequality for subharmonic function on minimal submanifolds yield
Theorem 6.4. (Xin-Yang)[17] Let x∈ M, R > 0 such that the image of DR(x) under the Gauss map lies in an open geodesic ball of radius
√ 2
4 π inGn,m. Then, there existsC1=C1(n, m), such that
(6.4) |B|2(x)≤C(n, m, p)R−(n+2p)( sup
DR(x)
h2)pVol(DR(x)), where p≥C1.
We also have
Theorem 6.5. (Xin-Yang)[18] Let x∈ M, R > 0 such that the image of DR(x) under the Gauss map lies in {P ∈ U ⊂ Gn,m : v(P) < 2}. Then, there exists C1=C1(n, m), such that
(6.5) |B|2s(x)≤C(n, s)R−(n+2s)( sup
DR(x)
˜h2)sVol(DR(x)),
for arbitrary s ≥ C1, where h˜2 is a composition of the Gauss map and the h = v2k(2−v)−2k withk= 34+2s1
Remark 6.1. For general submanifolds we can also carry out those curvature es- timates [16].
7 Some results on mean curvature flow
Consider the deformation of M under the mean curvature flow, i.e. ∃ a one- parameter family Ft =F(·, t) of immersionsFt: M → Rm+n with corresponding imagesMt=Ft(M) such that
d
d tF(x, t) =H(x, t), x∈M F(x,0) =F(x),
whereH(x, t) is the mean curvature vector ofMtatF(x, t).
For codimension one case, there are many deep results by G. Huisken among them Ecker-Huisken proved that
Theorem 7.1. If the initial hypersurfaceM0⊂Rm+1 is an entire graph with linear growth, then MCF equation has smooth long time solution.
It is natural to study the mean curvature flow of higher codimension. In recent years some works have been done: K. Smoczyk, M.T. Wang, J.Chen, J.Li.
• linear growth by E-H bounded gradient of the defined function by Moser.
Their geometric meaning is the Gauss image contained in open hemisphere.
We prove the so-called confinable property in higher codimension.
Theorem 7.2. (Xin)[15] If the Gauss image of the initial submanifoldM is con- tained in a geodesic ball of the radiusR <
√2
4 πinGm,n, then the Gauss images of all the submanifolds under the MCF are also contained in the same geodesic ball.
Let
h= 1 +ε−cos(√ 2ρ),
where ρ is the distance function from a point inGm,n, ε > 0 is a fixed constant.
Whenρ <
√ 2
4 π,his convex. For simplicityh◦γis denoted by h1.
Theorem 7.3. (Xin)[15] Let M be a complete m−submanifold in Rm+n with bounded curvature. Suppose that the image under the Gauss map from M into Gm,n lies in a geodesic ball of radius R <
√ 2
12π. If Mt is a smooth solution of the MCF equation, then there is the following estimate
(7.1) sup
Mt
|B|2hq1≤sup
M0
|B|2hq1,
whereq is a fixed constant depending onR0.
Theorem 7.4. (Xin)[15] LetF :M →Rm+n be a completem−submanifold which has bounded curvature. Suppose that the image under the Gauss map fromM into Gm,n lies in a geodesic ball of radius R <
√ 2
12π. Then the evolution equation of mean curvature flow has long time smooth solution.
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