˖ڍመڙጲ
http://www.paper.edu.cn
Remark that RHS of (0.1) dose not depend on the weight of the measure µ . Theorem
0.1 is a consequence of Theorem 3.6, a more general result on RCD
∗
-spaces. In the case of a
smooth Riemannian manifold (M, g) of n-dimension with the Riemannian volume µ := vol
g
,
the relation (0.1) recovers the classical Weyl law.
Let us look at the case of an n(> 2)-dimensional Alexandrov space (X, d) with the Haus-
dorff measure H
n
of n-dimension, with curvature > k for some k ∈ R. It was proved that
(X, d, H
n
) satisfies RCD
∗
((n − 1)k, n). From Theorem 0.1, we have the following consequence.
Corollary 0.2. Let Ω be a bounded domain in an n-dimensional Alexandrov space (X, d, H
n
).
Then we have the Weyl law
lim
λ→∞
N
Ω
(λ)
λ
n/2
=
ω
n
· H
n
(Ω)
(2π)
n
. (0.2)
If (X, d, µ) is a Gromov-Hausdoff limit space of a sequence of pointed Riemannian manifolds
(M
j
, g
j
, p
j
) with
Ric
M
j
> K, dim(M
j
) = n, vol
g
j
(B
1
(p
j
)) > v
0
> 0,
then the Weyl law (0.2) still holds. This case was proved by Ding in [17].
Recalling that a proof of the Weyl’s law on smooth setting, a key ingredient is that one can
deduce a uniformly small time asymptotic behaviour of heat trace H(t, x, x) from the parametrix
of heat kernels. However, the construction of the parametrix on smooth manifolds does not
work on singular metric measure spaces. To deal with this lack of the parametrix, we shall prove
the (locally) uniform convergce of Dirichlet heat kernels along a pointed measured Gromov-
Hausdorff converging sequence of metric measure spaces, which is similar as in [17, 18, 19].
As a byproduct, we also show a convergence of Dirichlet eigenvalues on a sequence of
RCD
∗
(K, N )-spaces, which is of independent interesting.
To consider a local Dirichlet problem under the Gromov-Hausdorff convergence, it is nat-
ural to assume that not only the balls B
R
(p
j
) converge to the limit ball B
R
(p
∞
), and also the
corresponding boundaries converge to the boundary of the limit ball. Precisely, let us introduce
the Gromov-Hausdorff convergence for pairs as follows.
Definition 0.3. Let (M, d
M
), (X, d
X
) be two compact metric spaces and let subsets A ⊂ M
and B ⊂ X. Given ν > 0, a map f : M → X is called a ν-GH-approximation from the pair
(M, A) to pair (X, B) if it is a ν-GH-approximation from M to X and also if
f(A) ⊂ U
ν
(B), B ⊂ U
ν
f(A)
,
where U
ν
(B) := {x ∈ X : d(x, B) < ν} is the ν-neighborhood of B in X.
Let (M
j
, d
j
) be a sequence of compact metric spaces and subsets A
j
⊂ M
j
. It is called
that the pairs (M
j
, A
j
) converge to a pair ( X, B) in the sense of Gromov-Hausdorff if there
exists a sequence of ν
j
-GH-approximations from the pair (M
j
, A
j
) to pair (X, B) with ν
j
→ 0
as j → ∞.
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