(School of Mathematics and Statistics, Henan University, Kaifeng 475004, China)

Abstract: In this paper, we introduce a kind of submanifold called translating solitons with density,and obtain two variational formulas for it,and show some geometric quantities of it.

Keywords: Translating solitons with density; Variational formula; Mean curvature flow

§1. Introduction

LetX:Mn →Rm+nbe an isometric immersion from ann−dimensional oriented Riemann manifoldMto the Euclidean space Rm+n.Let us consider the mean curvature flow for a submanifoldMin Rm+n.Namely, consider a one-parameter famillyXt=X(·,t) of immersionsXt:M →Rm+nwith corresponding imagesMt=Xt(M) such that

for any (x,t)∈M×[0,T),whereH(x,t) is the mean curvature vector ofMtatX(x,t) in Rm+n.

There is a special class of solutions to (1.1), called translating solitons (abbreviated by translators) of mean curvature flow. A submanifoldX:Mn →Rm+nis called traslator, if it satisifies

Here,Vis a constant unit vector in Rm+nandV Nis the normal projection ofVto the normal bundle ofMin Rm+n.The translator gives an eternal solutionX(t)=X+tVto equation (1.1).The translators play an important role in the study of mean curvature flow. They are not only special solutions to the mean curvature flow equations, but they also often occur as Type-IIsigularitiy of a mean curvature flow, see [1,2,4–6,9,12,13,15,16].

On the other side, an-dimension manifold with densityis a manifoldwith a metricand a functionwhere on anyk−dimensional submanifoldPof(1≤k ≤n),we consider the metricginduced bybut instead of the canonical volume elementassociated to the metricg, we use the volume elementinduced by the densityψ.The volume associated to the densityis called theψ−volume

Manifolds with density are being actively studied in many contexts. We refer to [10] for a short history in the context of mean curvature flow. Further more, mean curvature flow with density also attracts a lot of attentions, for more details, we refer to see [7,8,10,11,14].

In this paper, we’d like to consider the manifoldX:Mn →Rm+nsatisfying

Here,His the mean curvature vector ofM, Vis a constant vector with unit length in Rm+nandV Nis the normal projection ofVto the normal bundle ofMin Rm+n,and we call such manifolds translating solitons with density (abbreviated by translators with density).

To show the geometric quantity of the translator with density, firstly we regard Rm+nas a manifold with density, whose density function ishereVis a unit constant vector.Then theψ−volume of the submanifoldMin Rm+nis

wheredµis the volume ofM, induced from the ambient Euclidean metric.

Secondly, we define a conformally falt metricand obtain a Riemannian manifold, which is denoted byThenFII(M) is also the volume function ofMin

Now considering the variation of theψ−volume, we can obtain the following first variational formula for translator with density.

Theorem 1.1.(The first variational formula) Let X:M →Rm+n be a isometric immersed translator with density, let ft:M →Rm+n, |t|<ε, be a smooth family of immersions satisfying f0(M)=M, then we have

Here,to be the variational vector field alongft,andft(M)=Mt.

Remark 1.1.The translator with density M satisfies the Euler-Lagrangian equation of the ψ−volume (1.4).

Remark 1.2.The translator with density M can be regarded as a minimal submanifold in

Further more, by using the method from the minimal surface theory as in [18], we also can obtain the second variational formula for translator with density of codimension one.

But before we show the second variational formula, we introduce a linear operator onMin a similar manner of the drift-Laplacian on the self-shrinkers by Colding and Mincozzi [3],

it can be shown thatLIIis self-adjoint with respect to the measure

When translator with densityMis an oriented hypersurface, choose the variational vector field

whereνis the unit normal vector field ofMin Rn+1andφis any smooth function onMwith compact support. Then we have

Theorem 1.2.(The second variational formula)

The paper is orgnized as follows: in Section 2, we introduce some preliminaries of Remannian geometry; in Section 3, we give the proof of the upper two theorems.

§2. Preliminaries

In this section, we introduce some preliminaries of Riemannian geometry, for more details,we refer to see [17].

For eachp∈M,the tangent spacecan be decomposed to a direct sum ofTpMand its orthogonal complementNpMinSuch a decomposition is differentiable. So that we have an orthogonal decomposition of the tangent bundlealongM

Let (···)Tand (···)Ndenote the orthogonal projections into the tangent bundleTMand the normal bundleNMrespectively.

Define

forV,W ∈Γ(TM).We callBto be seconde fundamental form ofMin.

Taking the trace ofBgives the mean curvature vectorHofMinand

where{ei}is a local orthonormal frame field ofM.

IfH ≡0,thenMis called a minimal submanifold in

Further more, we can define the curvature tensorRX Y ZandRX Yµ, corresponding to the connections in the tangent bundle and the normal bundle respectively:

whereX,Y,Zare tangent vector fields,µis a normal vector field.

§3. Variational formulas for translating solitons with density

In this section, we can derive the variational formulas for translators with density from the minimal surface theory.

Firstly, we’d derive the first variation of the weight volume of the translator.

For the second term, we have

For the upper first term,

On the other side, we have

So

It leads to

This completes the proof of Theorem (1.1).

Now we shall derive the second variational formula from the minimal surface theory. Here we only consider the oriented codimension one case for possible applications.

When translator with densityMis an oriented hypersurface, choose the variational vector field

whereνis the unit normal vector field ofMin Rn+1andφis any smooth function onMwith compact support. Hence the above first variational formula becomes

with unit normal vectorνttoft(M) in Rn+1.

Let{e1,···,en}be a local orthonormal frame fields onMwith∇eiej=0 at the considered point.Denoteεibyεi=(ft)∗eiand

Att=0,we have

and then

It follows that

On the other hand,

However,

so whent=0,

also whent=0,

So combining (3.5) and the above equation,

Combining (3.4),

So combining (3.5) (3.6) and (3.7) we have

Combining(1.6), we can obtain

This is the second variation formula for oriented translating soliton with density of codimension one in Euclidean space.

Acknowledgements

The author would like to express his sincere gratitude to professors Li Ma from University of Science and technology in Beijing for his valuable suggestions and comments.