A Nonlinear Inequality Arising in Geometry and Calabi-Bernstein Type Problems
© A. Romero and R. M. Rubio. 2010
Received: 28 May 2010
Accepted: 18 September 2010
Published: 20 September 2010
A characterization for the entire solutions of a nonlinear inequality, which has a natural interpretation in terms of certain nonflat Robertson-Walker spacetimes, is given. As an application, new Calabi-Bernstein type problems are solved.
and the function , given by (1.1), is the mean curvature with respect to for the spacelike graph of (see Section 3 for details). Note that if (constant) then (1.1) reduces to , which is the mean curvature of the spacelike surface of defined by (it is called a spacelike slice). Thus, formula (1.1), with constant, and the constraint , constitute the constant mean curvature (CMC) spacelike graph equation in . Note that the constraint involving the length of the gradient of implies that the partial differential equation is elliptic. In a special case where , and , that is, when is the Lorentz-Minkowski spacetime, there are many entire (i.e., defined on all ) solutions of the CMC spacelike graph equation . This suggests that, when dealing with uniqueness results of entire solutions of the CMC spacelike graph equation in RW spacetimes, a stronger assumption than is needed (see below).
The geometric meaning of (I.2) is that the graph of is spacelike and . Moreover, (I.1) means that at the point of the graph of corresponding to , the absolute value of the mean curvature, is at most the absolute value of the mean curvature of the graph of the constant function , where . Note that we only suppose here a natural comparison inequality between two mean curvature quantities, but we don't require constant. Along the paper, inequality will mean inequality (I.1) with additional assumption (I.2).
It is clear that the constant functions are entire solutions of inequality . Our main aim in this paper is to state a converse under a suitable assumption on the warping function . In order to do that, we will work directly on spacelike surfaces instead of spacelike graphs. Recall that a spacelike surface is locally a spacelike graph and this holds globally under some extra topological hypotheses [2, Section ]. Our main tool is a local integral estimation of the squared length of the gradient of the restriction of the warping function on a spacelike surface. If is not locally constant (then, is said to be proper) and (which has an interesting curvature interpretation called the timelike convergence condition (TCC)), we first prove (Theorem 4.2).
Recall that a (general) noncompact 2-dimensional Riemannian manifold is parabolic if and only if as [3, Section ]. On the other hand, the Gauss curvature of the spacelike surface is nonnegative whereas the TCC and inequality hold true (see Section 3.2). Thus, using a well-known result by Ahlfors and Blanc-Fiala-Huber , we obtain that if is complete then it is parabolic. Therefore, approaches infinity for a fixed arbitrary point and a fixed , obtaining that is constant on . Since the RW spacetime is proper, this implies that must be a spacelike slice with . Thus, the first application of Theorem 4.2 is to reprove uniqueness result [5, Theorem ] with a local and different approach (Corollary 4.3).
It should be noted that inequality for assumed in Theorem 4.2 holds in a natural way under some suitable hypotheses on each complete CMC spacelike surface that lies between two spacelike slices [5, Section ]. However, note that we are not assuming here that is constant. In fact, Theorem 4.2 provides with several uniqueness results for complete spacelike surfaces whose constant mean curvature is only bounded (Corollaries 4.6 and 4.7).
Returning to our main aim, recall that an entire spacelike graph in an RW spacetime with fiber cannot be complete, in general (see, e.g., ). However, a graph of an entire function which satisfies (I.2) must be complete (Section 4). Therefore, as an application of the previous result we obtain the following uniqueness results in the nonparametric case (Theorems 4.8 and 4.9)
Finally, observe that inequality is trivially true in the maximal case; that is, for . Hence, our results contain new proofs of well-known Calabi-Bernstein type results (see [7, Theorem ]).
for any , where denotes the Levi-Civita connection of the metric (1.2), [1, Proposition ]. Thus, is conformal with , and its metrically equivalent 1-form is closed.
that is, such that . This curvature condition is the mathematical translation that gravity, on average, attracts and, on 4-dimensional spacetimes, holds whenever the metric tensor satisfies the Einstein equation (with zero cosmological constant) . We will also consider on the stronger condition , for any timelike tangent vector . When this holds, we will say that the TCC is strict on . Let us remark that, on 4-dimensional spacetimes, this curvature assumption indicates the presence of nonvanishing matter fields .
that is, which satisfies . A clear continuity argument shows that the TCC implies the NCC (on any -dimensional Lorentzian manifold). Note that any Einstein Lorentzian manifold (in particular, a Lorentzian space form) always satisfies the NCC.
In the case that is an RW spacetime with fiber , and making use again of [1, Corollary ], we can express the previous curvature conditions in terms of the warping function. Thus, obeys the TCC if and only if , the TCC strict if and only if and the NCC is equivalent to . It is easy to see that if there exists with , then the NCC implies the TCC. Moreover, if and there exists such that , then this zero of is unique and .
3.1. The Restriction of the Warping Function on a Spacelike Surface
Let be a (connected) spacelike surface in ; that is, is an immersion and it induces a Riemannian metric on the (2-dimensional) manifold from the Lorentzian metric (1.2). It should be noted that any spacelike surface in is orientable and noncompact . We represent the induced metric with the same symbol as the metric (1.2) does. The unitary timelike vector field allows us to consider as the only, globally defined, unitary timelike normal vector field on in the same time orientation of . Thus, from the wrong way Cauchy-Schwarz inequality, (see [1, Proposition ], for instance) we have and at a point p if and only if . By spacelike slice we mean a spacelike surface such that is a constant. A spacelike surface is a spacelike slice if and only if it is orthogonal to or, equivalently, orthogonal to .
where , and the function , where is the shape operator associated to , is called the mean curvature of relative to . A spacelike surface with constant mean curvature is a critical point of the area functional under a certain volume constraint (see , for instance). A spacelike surface with is called maximal. Note that, with our choice of , the shape operator of the spacelike slice with is and its mean curvature is .
3.2. The Gauss Curvature of a Spacelike Surface
4. Main Results
If and ( ) denote geodesic balls centered at the point of a Riemannian manifold, we recall that is the capacity of the annulus , being the harmonic measure of (see [3, Section ] for instance). First of all, we recall the following technical result.
Lemma 4.1 (see [12, Lemma ]).
Now, we are in a position to prove the announced local integral estimation.
As a first application of Theorem 4.2 we reprove the following well-known uniqueness result, using a different approach.
Corollary 4.3 (see [5, Theorem ]).
As mentioned in Section 2, if obeys the NCC and there exists such that , then also obeys the TCC. On the other hand, any maximal surface in clearly satisfies (4.2), hence we reprove and extend (with a different approach) the parametric version of the Calabi-Bernstein type result [7, Corollary ].
From our assumptions, the TCC and , we have . Therefore, the mean curvature of the spacelike surface satisfies (4.2). Thus, Theorem 4.2 can be then claimed to conclude the integral estimation (4.3). The proof ends making in this formula.
Analogously we can state the following corollary.
With an analogous reasoning we obtain the following theorem.
Observe that Theorem 4.9 trivially holds true if is assumed to be identically zero. Therefore, Theorem 4.9 reproves the well-known uniqueness result for the maximal surface equation [7, Theorem ]. On the other hand, Theorem 4.8, partially extends [14, Theorem ].
The authors are thankful to the referees for their deep reading and making suggestions towards the improvement of this paper. This work was partially supported by the Spanish MEC-FEDER Grant MTM2007-60731 and the Junta de Andalucia Regional Grant P09-FQM-4496 with FEDER funds.
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