Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

Remark 2.1

From the facts about the forced MCF (2.2) introduced above, we know that the solutions to evolution equations for the forced MCF exist on a maximal time interval if the initial hypersurface is strictly convex. Therefore, in what follows, we always assume [ 0 , T ) ( T > 0 ) to be the maximal time interval for the evolving hypersurface M t .

Proposition 3.1

Let λ 1 ( t ) be the first nonzero closed eigenvalue of the Laplace operator on an n-dimensional compact and strictly convex hypersurface ( M t n , g ( t ) ) ( n ≥ 2 ) under the flow (2.2). Let u ( x , t ) be the normalized eigenfunction corresponding to λ 1 ( t ) , namely, Δ u = − λ 1 u and ∫ M t u 2 = 1 . Then we have

Proof

Under the same assumptions of the aforementioned proposition, Mao [9] obtained

Since the hypersurface M t ⊂ ℝ n + 1 , the Codazzi equation h i j k = h i k j holds for i , j , k ∈ { 1 , 2 , … , n } , so ∇ i h i j = ∇ j H . By integration by parts, the last term of (3.2) becomes

Inputting (3.3) into (3.2), we get the desired equality (3.1).□

Proposition 3.2

Let λ 1, p ( t ) be the first nonzero closed eigenvalue of the p-Laplace operator on an n-dimensional compact and strictly convex hypersurface ( M t n , g ( t ) ) ( n ≥ 2) under the forced flow (2.2). Assume that u ( x , t ) is any smooth function satisfying (2.7) such that at time t 0 ∈ [ 0 , T ) , u ( x , t 0 ) is the eigenfunction corresponding to λ 1, p ( t 0 ) . Let λ 1, p ( u , t ) be a smooth function defined by (2.6). Then at time t = t 0 , we have

Proof

Under the same assumptions of the aforementioned proposition, at time t = t 0 , Mao [9] obtained

Since the hypersurface M t ⊂ ℝ n + 1 , the Codazzi equation h i j k = h i k j holds for i , j , k ∈ { 1 , 2 , … , n } , so ∇ i h i j = ∇ j H . By integration by parts, the last term of (3.5) becomes

Inputting (3.6) into (3.5), we get the desired equality (3.4).□

Lemma 4.1

[14, Corollary 2.5(i)] Let M 0 be a compact, strictly convex hypersurface of dimension n ≥ 2 without boundary, smoothly embedded in ℝ n + 1 . If h i j > 0 on M 0 at t = 0 , then it remains so on 0 ≤ t < T under the forced MCF (2.2).

Remark 4.2

By Lemma 4.1, we know that the strict convexity is preserved under the powers of MCF and the forced MCF (2.2). Therefore, in what follows, the condition “strict convexity” imposed on the evolving hypersurface M t is feasible.

Lemma 4.3

[14, Corollary 2.5(ii)] Let M 0 be a compact, strictly convex hypersurface of dimension n ≥ 2 without boundary, smoothly embedded in ℝ n + 1 . If ε H g i j ≤ h i j ≤ β H g i j , and H > 0 at the beginning for some constants 0 < ε ≤ 1 n ≤ β < 1 , then it remains so under the forced MCF (2.2) on 0 ≤ t < T .

Remark 4.4

Similar conclusions about the preservation of the pinching conditions on h i j under the forced MCF (2.2) can be found in [9, Lemma 4.3] where the pinching condition on h i j is actually an almost-umbilical condition as [9] says. In addition, once we let ε = β = 1 n in Lemma 4.3, then we know that the property of being totally umbilical is preserved under the forced MCF (2.2) if the initial closed hypersurface is strictly convex. Since a well-known result states that a totally umbilical hypersurface of ℝ n + 1 which is not totally geodesic is a round sphere, it would be very interesting to discuss how a hypersurface with an almost-umbilical condition is close to a round sphere (see [9, Remark 1.2] for the discussion). In general, “strictly convex” means that the second fundamental form h i j > 0 , which implies that H > 0 , namely, mean convex. When the hypersurface is totally umbilical, then h i j = 1 n H g i j , so the convexity is equivalent to mean convexity in this case.

Theorem 4.5

Let λ ˆ 1 ( t ) , λ ˜ 1, p ( t ) ( p ∈ ( 1 , ∞ ) ,   p ≠ 2 ) be the first nonzero closed eigenvalue of the Laplace operator and the p-Laplace operator on an n-dimensional compact and strictly convex hypersurface ( M t n , g ( t ) ) ( n ≥ 2 ) which evolves under the forced MCF (2.2), t ∈ [ 0 , T ) , respectively. Denote by H max ( 0 ) and H min ( 0 ) the maximum and minimum values of the mean curvature of the initial hypersurface M 0 . If the initial hypersurface M 0 satisfies the pinching condition

for some constants 0 < ε ≤ 1 n ≤ β < 1 , then we have that

1. If κ ( t ) ≥ ( 1 + 2 β ) ψ 2 ( t ) − σ 2 ( t ) 2 for t ∈ [ 0 , T ) , then λ ˆ 1 ( t ) is nonincreasing under the forced MCF (2.2) for t ∈ [ 0 , T ) . Furthermore, e ∫ 0 t ( 2 κ ( τ ) − ( 1 + 2 β ) ψ 2 ( τ ) + σ 2 ( τ ) ) d τ λ ˆ 1 ( t ) is nonincreasing under the forced MCF (2.2) for t ∈ [ 0 , T ) .

2. If κ ( t ) ≤ ( 1 + 2 ε ) σ 2 ( t ) − ψ 2 ( t ) 2 for t ∈ [ 0 , T ) , then λ ˆ 1 ( t ) is nondecreasing under the forced MCF (2.2) for t ∈ [ 0 , T ) . Furthermore, e ∫ 0 t ( 2 κ ( τ ) − ( 1 + 2 ε ) σ 2 ( τ ) + ψ 2 ( τ ) ) d τ λ ˆ 1 ( t ) is nondecreasing under the forced MCF (2.2) for t ∈ [ 0 , T ) .

3. If κ ( t ) ≥ ( 1 + p β ) ψ 2 ( t ) − σ 2 ( t ) p for t ∈ [ 0 , T ) , then λ ˜ 1, p ( t ) is nonincreasing and differentiable almost everywhere under the forced MCF (2.2) for t ∈ [ 0 , T ) .

4. If κ ( t ) ≤ ( 1 + p ε ) σ 2 ( t ) − ψ 2 ( t ) p for t ∈ [ 0 , T ) , then λ ˜ 1 , p ( t ) is nondecreasing and differentiable almost everywhere under the forced MCF (2.2) for t ∈ [ 0 , T ) , where

Proof

For the forced MCF (2.2), since the initial hypersurface M 0 satisfies the pinching condition (4.1), by Lemma (4.3), we know that the evolving hypersurface M t also satisfies the pinching condition (4.1), that is, ε H g i j ≤ h i j ≤ β H g i j , for some constants 0 < ε ≤ 1 n ≤ β < 1 , for any t ∈ [ 0 , T ) . Locally, we can choose an orthogonal frame { e i } i = 1 n such that g i j = δ i j ,   i ,   j = 1 , 2 , … , n , then let h i j = μ i δ i j ,   i ,   j = 1 , 2 , … , n , where μ i ( x , t ) is the principal curvature at F ( x , t ) , so for every i ∈ { 1 , 2 , … , n } ,

Since the evolving hypersurface M t is strictly convex, by (4.2) and the Cauchy-Schwartz inequality, we have

that is,

As Mao [9, Theorem 5.1] does, in view of (4.4), by applying the maximum principle to (2.5), we obtain the following estimates for H ( x , t ) of the general n-dimensional compact and strictly convex hypersurface evolving along the forced MCF (2.2), that is,

for t ∈ [ 0 , T ) , where

For the case of the Laplace operator, let u ( x , t ) be the normalized eigenfunction corresponding to λ ˆ 1 ( t ) , namely, Δ u = − λ ˆ 1 u and ∫ M t u 2 = 1 . By (3.1) in Proposition 3.1, we have

and

then d d t λ ˆ 1 ( t ) ≤ 0 provided κ ( t ) ≥ ( 1 + 2 β ) ψ 2 ( t ) − σ 2 ( t ) 2 for t ∈ [ 0 , T ) and d d t λ ˆ 1 ( t ) ≥ 0 provided κ ( t ) ≤ ( 1 + 2 ε ) σ 2 ( t ) − ψ 2 ( t ) 2 for t ∈ [ 0 , T ) . Then we conclude that λ ˆ 1 ( t ) is nonincreasing under the forced MCF (2.2) provided κ ( t ) ≥ ( 1 + 2 β ) ψ 2 ( t ) − σ 2 ( t ) 2 for t ∈ [ 0 , T ) and λ ˆ 1 ( t ) is nondecreasing under the forced MCF (2.2) provided κ ( t ) ≤ ( 1 + 2 ε ) σ 2 ( t ) − ψ 2 ( t ) 2 for t ∈ [ 0 , T ) . Furthermore, from (4.6) and (4.7), we infer that e ∫ 0 t ( 2 κ ( τ ) − ( 1 + 2 β ) ψ 2 ( τ ) + σ 2 ( τ ) ) d τ λ ˆ 1 ( t ) is nonincreasing under the forced MCF (2.2) for t ∈ [ 0 , T ) and e ∫ 0 t ( 2 κ ( τ ) − ( 1 + 2 ε ) σ 2 ( τ ) + ψ 2 ( τ ) ) d τ λ ˆ 1 ( t ) is nondecreasing under the forced MCF (2.2) for t ∈ [ 0 , T ) .

Now for the case of the p-Laplace operator, we assume that u ( x , t ) is any smooth function satisfying (2.7) such that at time t 0 ∈ [ 0 , T ) , u ( x , t 0 ) is the eigenfunction corresponding to λ ˜ 1, p ( t 0 ) . Let λ ˜ 1 , p ( u , t ) be a smooth function defined by (2.6). Similar to the aforementioned process, by (3.4) in Proposition (3.2), at time t = t 0 , we obtain

and

then d d t λ ˜ 1 , p ( u , t ) | t = t 0 ≤ 0 provided κ ( t ) ≥ ( 1 + p β ) ψ 2 ( t ) − σ 2 ( t ) p for t ∈ [ 0 , T ) and d d t λ ˜ 1 , p ( u , t ) | t = t 0 ≥ 0 , provided κ ( t ) ≤ ( 1 + p ε ) σ 2 ( t ) − ψ 2 ( t ) p for t ∈ [0, T ) . Since λ ˜ 1 , p ( u , t ) is a smooth function with respect to t-variable, we have d d t λ ˜ 1 , p ( u , t ) ≤ 0 in any sufficiently small neighbourhood of t 0 provided κ ( t ) ≥ ( 1 + p β ) ψ 2 ( t ) − σ 2 ( t ) p for t ∈ [0, T ) . Integrating the inequality with respect to t on a sufficiently small time interval [ t 0 , t 1 ] ( t 0 ≤ t 1 ) , we get λ ˜ 1 , p ( u ( x , t 0 ) , t 0 ) ≥ λ ˜ 1 , p ( u ( x , t 1 ) , t 1 ) . Note that λ ˜ 1 , p ( u ( x , t 0 ) , t 0 ) = λ ˜ 1 , p ( t 0 ) , λ ˜ 1 , p ( u ( x , t 1 ) , t 1 ) ≥ λ ˜ 1 , p ( t 1 ) , therefore, λ ˜ 1 , p ( t 0 ) ≥ λ ˜ 1 , p ( t 1 ) for any t 1 ≥ t 0 . As t 0 ∈ [ 0 , T ) is arbitrary, then we conclude that λ ˜ 1 , p ( t ) is nonincreasing under the forced MCF (2.2) provided κ ( t ) ≥ ( 1 + p β ) ψ 2 ( t ) − σ 2 ( t ) p for t ∈ [0, T ) . By the similar process, we can obtain λ ˜ 1 , p ( t ) is nondecreasing under the forced MCF (2.2) provided κ ( t ) ≤ ( 1 + p ε ) σ 2 ( t ) − ψ 2 ( t ) p for t ∈ [0, T ) . The assertion that λ ˜ 1, p ( t ) is differentiable almost everywhere for t ∈ [ 0 , T ) can be derived by the classical Lebesgue’s theorem which states that a monotonous continuous function is differentiable almost everywhere.□

Remark 4.6

In terms of Theorem 4.5, when β = 1 n , by (4.4), we know | A | 2 = H 2 n , so M 0 is totally umbilical. By Remark 4.4 and the fact that M 0 is strictly convex and closed, so M 0 is a round sphere, H max ( 0 ) = H min ( 0 ) , then σ ( t ) = ψ ( t ) by their definition. Under additional condition that κ ( t ) ≤ 0 , for t ∈ [ 0 , T ) , obviously κ ( t ) ≤ 0 < ( 1 + 2 ε ) σ 2 ( t ) − ψ 2 ( t ) 2 for any t ∈ [ 0 , T ) and also κ ( t ) ≤ 0 < ( 1 + p ε ) σ 2 ( t ) − ψ 2 ( t ) p for any t ∈ [ 0 , T ) . Particularly, when κ ( t ) ≡ 0 , for t ∈ [ 0 , T ) , the forced MCF (2.2) reduces to the classical MCF, then from the proof of Theorem 4.5, we conclude that the first eigenvalue of the Laplace operator and the p-Laplace operator are strictly increasing under the forced MCF (2.2) if the initial hypersurface is a round sphere and κ ( t ) ≤ 0 , for t ∈ [ 0 , T ) . Regarding the normalized MCF case which corresponds to the case when κ ( t ) = h ( t ) n , where h ( t ) = ∫ M t H 2 d μ t ∫ M t d μ t , by Theorem 4.5, we infer that the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator are constants under the normalized MCF if the initial hypersurface is a round sphere. In fact, we know that the evolving hypersurface M t preserves the property of being totally umbilical, so if the initial hypersurface is a round sphere, then the evolving hypersurface M t is also a round sphere. As we know, the evolving hypersurface M t preserves the total area, so the evolving hypersurface M t as a round sphere has the same radius as the initial round sphere, then the first eigenvalue is a constant, for t ∈ [0, T ) . Moreover, the second assertion of (ii) in Theorem 4.5 coincides with [6, Theorem 1.2] ( k = 1 ) when κ ( t ) ≡ 0 , for t ∈ [ 0 , T ) .

Theorem 4.7

Let λ ˆ 1 ( t ) be the first nonzero closed eigenvalue of the Laplace operator on an n-dimensional compact and strictly convex hypersurface ( M t n , g ( t ) ) ( n ≥ 2 ) which evolves under the forced MCF (2.2), t ∈ [0, T ) . Denote by H max ( 0 ) and H min ( 0 ) the maximum and minimum values of the mean curvature of the initial hypersurface M 0 . If the initial hypersurface M 0 satisfies the pinching condition (4.1) for some constants 0 < ε ≤ 1 n ≤ β < 1 , then we have that

1. If κ ( t ) ≤ σ 2 ( t ) + ( 2 ε − 1 ) ψ 2 ( t ) 2 for t ∈ [0, T ) , and 1 n ≤ β ≤ 1 2 , then λ ˆ 1 ( t ) is nondecreasing under the forced MCF (2.2) for t ∈ [0, T ) . Furthermore, e ∫ 0 t ( 2 κ ( τ ) − σ 2 ( τ ) − ( 2 ε − 1 ) ψ 2 ( τ ) ) d τ λ ˆ 1 ( t ) is nondecreasing under the forced MCF (2.2) for t ∈ [ 0 , T ) .

2. If κ ( t ) ≥ ψ 2 ( t ) + ( 2 β − 1 ) σ 2 ( t ) 2 for t ∈ [0, T ) and 1 n ≤ β ≤ 1 2 , then λ ˆ 1 ( t ) is nonincreasing under the forced MCF (2.2) for t ∈ [ 0 , T ) . Furthermore, e ∫ 0 t ( 2 κ ( τ ) − ψ 2 ( τ ) − ( 2 β − 1 ) σ 2 ( τ ) ) d τ λ ˆ 1 ( t ) is nonincreasing under the forced MCF (2.2) for t ∈ [0, T ) .

3. If κ ( t ) ≥ β ψ 2 ( t ) for t ∈ [ 0 , T ) and 1 2 ≤ β < 1 , then λ ˆ 1 ( t ) is nonincreasing under the forced MCF (2.2) for t ∈ [0, T ) . Furthermore, e ∫ 0 t ( 2 κ ( τ ) − 2 β ψ 2 ( τ ) ) d τ λ ˆ 1 ( t ) is nonincreasing under the forced MCF (2.2) for t ∈ [0, T ) , where σ ( t ) , ψ ( t ) are defined as in Theorem 4.5.

Proof

The proof is similar to that of Theorem 4.5. Let u ( x , t ) be the normalized eigenfunction corresponding to λ ˆ 1 ( t ) , namely, Δ u = − λ ˆ 1 u and ∫ M t u 2 = 1 . Let’s deal with the following terms in (3.1):

Obviously,

As is shown in the proof of Theorem 4.5, the evolving hypersurface M t satisfies the pinching condition (4.1) for some constants 0 < ε ≤ 1 n ≤ β < 1 , then 0 < ε H g i j ≤ h i j ≤ β H g i j , therefore, if 1 n ≤ β ≤ 1 2 , by (4.5), (4.10), we have

Inputting (4.11) into (3.1), by (4.5), we have

and

If 1 2 ≤ β ≤ 1 , by (4.5), (4.10), we have

Inputting (4.14) into (3.1), by (4.5), we have

then by (4.12), (4.13), if 1 n ≤ β < 1 2 , we conclude that λ ˆ 1 ( t ) is nondecreasing under the forced MCF (2.2) provided κ ( t ) ≤ σ 2 ( t ) + ( 2 ε − 1 ) ψ 2 ( t ) 2 for t ∈ [ 0 , T ) , and e ∫ 0 t ( 2 κ ( τ ) − σ 2 ( τ ) − ( 2 ε − 1 ) ψ 2 ( τ ) ) d τ λ ˆ 1 ( t ) is nondecreasing under the forced MCF (2.2) for t ∈ [0, T ) , and also λ ˆ 1 ( t ) is nonincreasing under the forced MCF (2.2) provided κ ( t ) ≥ ψ 2 ( t ) + ( 2 β − 1 ) σ 2 ( t ) 2 for t ∈ [0, T ) , and e ∫ 0 t ( 2 κ ( τ ) − ψ 2 ( τ ) − ( 2 β − 1 ) σ 2 ( τ ) ) d τ λ ˆ 1 ( t ) is nonincreasing under the forced MCF (2.2) for t ∈ [0, T ) . If 1 2 ≤ β < 1 , by (4.15), then we conclude that λ ˆ 1 ( t ) is nonincreasing under the forced MCF (2.2) provided κ ( t ) ≥ β ψ 2 ( t ) , for t ∈ [ 0 , T ) , and e ∫ 0 t ( 2 κ ( τ ) − 2 β ψ 2 ( τ ) ) d τ λ ˆ 1 ( t ) is nonincreasing under the forced MCF (2.2) for t ∈ [ 0 , T ) .□