Pengfei Guanand Junfang Li

1Department of Mathematics and Statistics,McGill University,Montreal,Quebec H3A 0B9,Canada;

2Department of Mathematics,University of Alabama at Birmingham,Birmingham,Al 35294,USA.

Abstract.New types of hypersurface flows have been introduced recently with goals to establish isoperimetric type inequalities in geometry.These flows serve as efficient paths to achieve the optimal solutions to the problems of calculus of variations in geometric setting.The main idea is to use variational structures to develop hypersurface flows which are monotonic for the corresponding curvature integrals(including volume and surface area).These new geometric flows pose interesting but challenging PDE problems.Resolution of these problems have significant geometric implications.

Key words:Hypersurface curvature flows,geometric inequalities,quermassintegrals.

1 Introduction

It has been observed that the isoperimetric difference is decreasing along the curve shortening flow[9]

where κ is the curvature of the boundary and ν the outer normal.Let|?|and|??|be the area and perimeter of a bounded domain ??R2.Along the curve shortening flow,let ?(t)be the domain at time t.It follows from the Gauss-Bonnet Theorem and the Cauchy-Schwarz inequality that,the isoperimetric difference is monotonic decreasing(and strictly decreasing if ?(t)is not a round ball).The convergence of curve shorting flow(1.1)yields the classical isoperimetric inequality in R2.

If ?0?N is an optimal domain in a space N of the isoperimetric problem,then ??0is a hypersurface of constant mean curvature.The isoperimetric problem can be considered as a problem of calculus of variation:for given A,find a domain ? such that V(?)=|?|is of least volume among all domains in N with A(?)=|??|=A.We search for an effective path under volume constraint to achieve an optimal domain.For any variational vector field η,let fν be its normal component.Then

where g is the induced metric of the boundary??.The volume is preserved if and only if

That is,the normal component f is orthogonal to the kernel of?g.This is the case if and only if

for some Φ on ??.

One has freedom to pick any Φ.We would like to search Φ such that to ensure the monotonicity of the hypersurface area.Since ? may evolve,we look for Φ which is defined in N(or a region of N).

Let’s first consider N=Rn+1.For any bounded domain ??Rn+1with smooth boundary ??,let X denote the position vector of the boundary surface,and let|X|be the distance from the origin.In(1.2),we choose where h is the second fundamental form of ??.Denote σk(λ1,···,λn)to be the k-th elementary symmetric function defined in Rn,which can be extended to be defined in n×n symmetric matrices h=(hij).Denote

where the last inequality follows from Heintze-Karcher inequality see[3,20,25,26].Consequently,one can deduce the sharp isoperimetric inequality if the initial ?? is mean convex(this condition is not needed for flow(1.3))and if long time existence and exponential convergence of the flow can be proved,then

In contrast to flow(1.3),flow(1.9)is a normalized flow of the standard inverse mean curvature flow[11].

There are other hypersurface flows discussed in the literature related to geometric inequalities.Most of them(e.g.[18,19,30])are “normalized flows” with associated to certain geometric quantity,where there are non-local terms involved.The novelty of(1.3)is that we work directly on normalized flows.One advantage of this is that the C0estimate is immediate by the maximum principle.Another interesting feature is the monotonic properties of all the quermassintegrals along the flow.This special feature will be figured prominently in the rest discussion of this article.

2 Mean curvature type flows in space forms and warped product spaces

We want to generalize the argument discussed to solve the isoperimetric problem in general spaces.Let’s consider space forms.Recall the Gaussian normal coordinates for the metric of a space form

where dz2is the standard reduced metric of a unit sphere in Rn+1.The cases of φ(r)=sin ρ,ρ,sinh ρ yield metrics of Sn+1,Rn+1,and Hn+1respectively.There is a natural choice of function Φ which plays a similar role as|X|2/2 in the Euclidean space discussed in the previous section.Let

Since powers of volume of balls and powers of area of the boundary spheres do not always have simple algebraic relation in space forms except in Rn+1,instead of having a simple isoperimetric inequality as in Rn+1,we conclude an isoperimetric comparison inequality with geodesic balls by establishing long time existence and exponential convergence[15].If ? has the same volume of a geodesic ball B,

then the surface area of?? is not less than the area of the sphere

with equality holds if and only if ? is a geodesic ball.

Thus,the crucial monotonicity is held,

A solution to the isoperimetric problem for warped product spaces can be obtained as follows.Let S(r)be a level set of r and B(r)be the bounded domain enclosed by S(r)and S(r0).The volume of B(r)and surface area of S(r),both positive functions of r,are denoted as V(r)and A(r),respectively.Note that V=V(r)is strictly increasing function of r.Consider the single variable function ξ(x)that satisfies

Theorem 2.1([17]).Let ??Nn+1be a domain bounded by a smooth graphical hypersurface M and S(r0)with n≥2.Suppose?? is inside the regionwhere is is the base manifold.Suppose

The condition(φr)2?φrrφ ≤ K is necessary since this is equivalent to the condition of stability of slice{r=c}as a hypersurface of constant mean curvature.The condition 0≤(φr)2?φrrφ is imposed in[17]for the gradient estimates of PDE of the radial function.Note that for flows(1.3),(2.2)and(2.7),?? is not assumed to any convex assumptions.

3 Fully nonlinear flows and quermassintegrals in Rn+1

We consider hypersurface flows related to the quermassintegrals.In convex geometry,there is the notion of quermassintegrals.If ??Rn+1is a C2domain,the quermassintegrals can be expressed in terms of boundary curvature integrals:

This would immediately imply the sharp quermassintegral inequalities in convex geometry.In question is the longtime existence and convergence of flow(3.5).The major PDE problem for flow(3.5)is the curvature estimate(or C2estimates).To overcome this difficulty,we transform flow(3.5)to a parabolic PDE on Snin[16].Let ??Rn+1be a bounded strictly convex domain with smooth boundary.Let Wijbe its Weingarten curvature tensor.It is well-known that the boundary hypersurface can be parametrized by the support function of the inverse of its Gauss map if the domain is strictly convex.Namely,the support function u=u(ν),where ν ∈Snare the outward normal vector at points on the boundary.With this parametrization,the inverse of the Weingarten curvature has a simple form

where eijis the metric tensor of the standard unit sphere Sn.The induced metric of the hypersurface satisfies gij:=eklAikAjl.

One can show that the following parabolic PDE of the support function u is equivalent to flow(3.5)for convex domains,

Since by the Newton McLaurine inequality,

This yields that u is a super-harmonic function on Sn.So u is a constant.This implies the convergence of flow(3.6).

We will use a similar argument as in Proposition 5.5 of[15]to show that when t is large enough,different eigenvalues of Aijat the same point are comparable uniformly for arbitrary small e.

From(3.26)and(3.27),we have I0(s)≤0.By our regularity estimates,we have I(t)is uniformly bounded from above and below.Thus,

Since we have established uniform a priori estimates for all the derivatives of u for any order and also 00,there exists a large enough T0>0,such that for any t>T0,the Newton-MacLaurin difference can be arbitrarily small.Namely,

This completes the proof of the lemma.

Lemma 3.4 yields that Aijis positive definite with eigenvalues bounded from below and above when t large.With uniform convexity,when t is large,

At maximum point(t,x0)of|?u(t,x)|with ?u=(u1,0,···,0),since u1j=0 so that A11=u.We may assume Gijdiagonal,it follows from(3.17),

That is,|?u|2is convergent to 0 exponentially.One may also go back to the corresponding hypersurface flow(3.5)to deduce the exponential convergence.Aijis uniform definite when t large,thus flow(3.5)is uniformly parabolic.Then one may infer Proposition 3.1 in[16]to get the exponential convergence.

We do not know if the convexity of Aijis preserved along flow(3.6),this would imply the curvature estimates and longtime existence for flow(3.5).

4 Fully nonlinear flows in Hn+1and Sn+1

Let Nn+1(K)be a space form of constant sectional curvature K.There is corresponding notion of qumermassintegrals Ak(?)for convex domain ? in Nn+1(K)(e.g.,[27]).If the boundary?? is C2,it holds Cauchy-Cronfton formula,

4.1The Case Nn+1=Hn+1

The main problem here is preservation of starshapedness of flow(4.6),equivalently,the gradient estimate for the corresponding flow(4.7)or flow(4.8).Let

be the linearized operator of flow(4.8),where

since convexity implies gradient estimates.With that,the speed function is bounded from below.This yields that the convexity of hypersurfaces are preserved along flow(4.6).Hence,one has all the a priori estimates and convergence.

Theorem 4.1.Let M0be a radial graph of function ρ0over Snin Hn+1.Suppose either k=n?1,or Condition(4.15)is satisfied.Then flow(4.6)exists all time and convergent exponentially to a geodesic sphere.To be precise,solution γ(z,t)of(4.8)exists in interval[0,∞),and there exist a uniform constant α >0 which depends only on the initial graph,such that for any(z,t)∈Sn×[0,∞],

where the covariant derivatives are with respect to the spherical metric on Sn.

As a consequence,sharp isoperimetric inequality comparing An?1with all other Akfor convex domains in Hn+1can be proved.In two other special cases,we can manage to get around and obtain sharp quermassintegral inequalities.More specifically,we can compare the first and second quermassintegrals,A2and A1,with A0respectively.

In the case of h-convexity,full range of quermassintegral inequalities were obtained in[10,31]using contracting type of flows.Very recently,the results in[31]for h-convex domains in Hn+1were reproved using flow(4.6)directly in[21]by establishing that hconvexity is preserved along flow(4.6).The sharp relation between A2and A0was previously proved in[24]by a different method.

Recall the inverse mean curvature flow

and inverse curvature flow

studied in[12](see also[8]).

Lemma 4.1.Let M(t)be a smooth family of hypersurfaces.

We provide two proofs for the sharp geometric inequality between A1and A0for a star-shaped domain ??Hn+1with smooth boundary.Notice that if the boundary hypersurface satisfies the gradient bounds in Theorem 4.1,then the sharp inequality follows immediately from the long time existence and exponential convergence of flow(4.6)with k=1.

As a conclusion,we have the following sharp geometric inequalities.

Theorem 4.2.([4])Suppose ? is a bounded domain in Hn+1with smooth boundary.The following three results hold:

Case(1)in the theorem for l=2 was proved in[24]using different flow.We believe Condition(4.15)is redundant in above Theorem.We also refer[5]for the Minkowski inequality in the anti-de Sitter-Schwarzschild space.

Proof.Cases(2)and(3)follow from the longtime existence and convergence of flow(4.6)under Condition(4.15)or for k=n?1.We provide two proofs for l=1 of Case(1).

Proof 1.We wixll combine flow(4.6)and inverse mean curvature flow to complete the proof.Let M(t)be a solution to the inverse mean curvature flow(4.19)with initial condition M(0)=??.By Gerhardt’s estimate for the radial function ρ(t),there exists a large enough T>>0,such that

4.2 The case Nn+1=

4.3 Conclusion remarks

Flow approach for geometric inequalities is not new,however the constraint hypersurface flows discussed here for isoperimetric problems are different from previous works.The guiding idea is to use variational properties of the concerned geometric functionals F and G along variational field η =fν,

Acknowledgments

Part of this article(proof in Section 3)was completed while the xfirst author was visiting RIMS,Kyoto University.He would like to thank Professor Kaoru Ono for hosting him and thank RIMS for the warm hospitality.