Heat content in non-compact Riemannian manifolds

Let $\Omega$ be an open set in a complete, smooth, non-compact, $m$-dimensional Riemannian manifold $M$ without boundary, where $M$ satisfies a two-sided Li-Yau gaussian heat kernel bound. It is shown that if $\Omega$ has infinite measure, and if $\Omega$ has finite heat content $H_{\Omega}(T)$ for some $T>0$, then $H_{\Omega}(t)<\infty$ for all $t>0$. Comparable two-sided bounds for $H_{\Omega}(t)$ are obtained for such $\Omega$.


Introduction
Let (M, g) be a complete, smooth, non-compact, m-dimensional Riemannian manifold without boundary, and let ∆ be the Laplace-Beltrami operator acting in L 2 (M ). It is well known (see [5], [6], [10], [15]) that the heat equation ∆u(x; t) = ∂u(x; t) ∂t , x ∈ M, t > 0, (1.1) It was shown ( [2]) that if Ω is non-empty, bounded, and ∂Ω is of class C ∞ , and if (M, g) satisfies exactly one of the following three conditions: (i) M is compact and without boundary, (ii) (M, g) = (R m , g e ) where g e is the usual Euclidean metric on R m , (iii) M is a compact submanifold of R m with smooth boundary and g = g e | M , then there exists a complete asymptotic series such that where J ∈ N is arbitrary, and where the β j : j = 0, 1, 2, . . . are locally computable geometric invariants. In particular, we have that where |Ω| is the measure of Ω, and Per(Ω) is the perimeter of Ω. For earlier results in the Euclidean setting we refer to [12], [13], [14], and subsequently to [1], [3], and [4].
Define u Ω : Ω × (0, ∞) → R by If Ω has infinite measure and |∂Ω| = 0, then the convergence is also in L 1 loc (M ) (Section 7.4 in [7]). In this paper we obtain bounds for the heat content in the case where Ω has possibly infinite measure or infinite perimeter, and where M satisfies the following condition.
There exists C ∈ [2, ∞) such that for all x ∈ M, y ∈ M, t > 0, R > 0, where B(x; R) = {y ∈ M : d(x, y) < R}, and d(x, y) denotes the geodesic distance between x and y.
It was shown independently in [8] and [9] that M satisfying a volume doubling property and a Poincaré inequality is equivalent to M satisfying a parabolic Harnack principle, and is also equivalent to the Li-Yau bound (1.6) above. See for example Theorem 5.4.12 in [10]. We included (1.7) in the definition of the constant C, even though the volume doubling property is implied by (1.6).
Hence u Ω , defined by (1.4), is the unique, bounded solution of (1.1) with initial condition (1.5) in the sense of L 1 loc (M ). Moreover stochastic completeness holds. That is for all x ∈ M, t > 0, M dy p M (x, y; t) = 1. (1.8) We refer to Chapter 9 in [6].
(iii) If (1.7) holds for all x ∈ M , R > 0 then We make the following.
Our main result is the following.
for all t > 0. (ii) If (1.11) holds for some t = T > 0, then for all t > 0, where . (1.13) If Ω has finite Lebesgue measure, then we define the heat loss of Ω in M at t by F Ω (t) = |Ω| − H Ω (t).
(1.14) We have that the heat loss t → F Ω (t) of Ω in M is increasing, concave, subadditive, and continuous. If Ω is bounded and ∂Ω is smooth, then, by (1.3), there exists an asymptotic series of which the first few coefficients are known explicitly. Theorem 1.3 below concerns the general situation |Ω| < ∞, and gives bounds in non-classical geometries where e.g. either Ω has infinite perimeter, and/or ∂Ω is not smooth. .
(1.16) This paper is organised as follows. In Section 2 we give the proofs of Theorem 1.2 and Theorem 1.3. In Section 3 we analyse an example of Ω in R m where precise analysis of H Ω (t) is possible.
Next suppose that 0 < t ≤ T . By (1.9), and (2.3) for t = T , we have that This completes the proof of the assertion in part (i).
(ii) Let n > 0, p ∈ Ω, R > 0, and Ω n = Ω ∩ B(p; n), and suppose that (1.11) holds for some t = T > 0. Then |Ω n | ≤ |B(p; n)| < ∞. Reversing the roles of x and y in (2.2) we have that for d(x, y) < R, We have that (2.5) Using (1.6) and (2.4), we see that To bound the second term in the right-hand side of (2.5), we note that Hence, where we have used (2.7), (1.9), the lower bound in (1.6), and (1.10). We now choose R 2 such that the coefficient of H Ωn (t) in the right-hand side of (2.8) is equal to 1 2 . That is Rearranging and bootstrapping gives, by (2.5)-(2.9), and the fact that (2.10) We choose t = t T such that R * = T , and take the limit n → ∞ in the righthand side of (2.10). This limit is finite by the hypothesis at the beginning of the proof. We conclude that × Ω dx µ Ω (x; T ) |B(x; T )| (2.11)

Analysis of an example
In this section we present the asymptotic analysis of H Ω (t) as t ↓ 0, of an open set Ω in M = R m consisting of disjoint balls with centres in Z m , and decreasing radii. Recall that p R m (x, y; t) = (4πt) −m/2 e −|x−y| 2 /(4t) . Let where (z i ) i∈N is an enumeration of Z m , and where r 1 ≥ r 2 ≥ . . . . Furthermore, let δ = 1 − 2r 1 > 0.
(3.2) Theorem 3.1 (ii) below asserts that if H Ω (t) < ∞ for all t > 0, and if (3.2) holds then the balls loose heat independently as t ↓ 0 up to a term exponentially small in t.
Below we consider four main regimes: 1 2m < α < 1 m , 1 m < α < 1 m−1 , 1 m−1 < α < 1 m−2 , and 1 m−2 < α. The latter regime is absent for m = 2. In the first regime Ω has infinite measure, and Theorem 1.2 (iii) gives the order of magnitude as t ↓ 0. This has been refined in (3.5)-(3.6) below. In the second regime Ω has infinite perimeter, and Theorem 1.3 gives the order of magnitude as t ↓ 0. This has been refined in (3.12)-(3.13) below. In the third and fourth regimes Ω has finite perimeter. Theorem 1.3 gives two-sided bounds of order t 1/2 . In (3.9) and (3.11) below we show that the perimeter term appears with the usual numerical constant. The remainder estimates depend on whether i∈N r m−2 i is infinite or finite. Furthermore there are several borderline cases: α = 1 m , 1 m−1 , 1 m−2 . They all involve logarithmic corrections in the heat content. We only analyse, as an example, the case α = 1 m−2 . The latter case is again absent for m = 2.
To prove part (ii) we note that the lower bound in (3.4) follows from the first inequality in (3.12). To prove the upper bound we observe that if x ∈ B(z i ; r i ), y ∈ B(z j ; r j ), i = j, then which gives the bound in (3.4).
Proof of Theorem 3.2. We first consider the case 1 2m < α < 1 m . By (3.4), it suffices to consider the sum in the left-hand side of (3.8) (3.14) A straightforward application of Tonelli's theorem gives the formulae under (3.5) and (3.6). To obtain a lower bound for the left-hand side of (3.14), we use the monotonicity of i → H B(0;ai −α ) (t) once more, and obtain that The last term in the right-hand side of (3.15) is bounded in absolute value by This completes the proof of the assertion under (3.5) and (3.6).
Consider the case 1 m < α < 1 m−1 . By (3.4), and scaling we have that In a similar way to the proof of (3.5),(3.6), we approximate the sum with respect to i by an integral.
3. Let f : R + → R + be increasing, and let g : Proof. We have that , .
To obtain an upper bound, we let J ∈ N, and note that by (3.4), The third term in the right-hand side of (3.24) is O(J 1−α(m−1) )t 1/2 . The fourth term in the right-hand side of (3.24) is O(J 1−α(m−2) )t. The choice J = ⌊t −1/(2α) ⌋ gives a remainder O(t (mα−1)/(2α) ) for the upper bound, and completes the proof of (3.9). Next consider the case α = 1 m−2 . The sum of the third and fourth terms in the right-hand side of (3.23) equals, up to constants, I −2/(m−2) +t log I. We now choose I = ⌊t −(m−2)/2 ⌋, and obtain the remainder in (3.10). Similarly, the sum of the third and fourth terms in the right-hand side of (3.24) is of order J −1/(m−2) t 1/2 + t log J. We now choose J = ⌊t −(m−2)/2 ⌋ to obtain the same remainder.